*“Freedom is the freedom to say that 2+2=4. If that is granted, all else follows” -George Orwell, Nineteen Eighty-four*

Occasionally in the course of human events it becomes necessary to have to explain something no one would ever have expected to have to defend. In the present moment, we find that circumstance to be the case and that thing to be that two and two do, in fact, make four. Further, it must be reasserted, against all reasonable expectation, that this claim about the sum of two and two being four is not merely some subjective determination or, more insidiously, an assertion of hegemonic power. So it is that such a need might arise in such a time in which irrational subjectivity becomes so desperate to defend and assert itself that no truth, no matter how simple or basic, can be considered safe from the ravages of people who have a vested ideological interest in its being wrong.

I have to confess responsibility for this bizarre moment, which in some sense might be one of the greater achievements of my life thus far. There’s an excellent case to be made that I have led a significant number of professionals who definitely should know otherwise—as effectively every six-year-old in a community with a school does—to dig deeply into tortured defenses of the proposition that two and two do not make four.

### What in the World…?

*“He wondered, as he had many times wondered before, whether he himself was a lunatic. Perhaps a lunatic was simply a minority of one. At one time it had been a sign of madness to believe that the earth goes round the sun; to-day, to believe that the past is inalterable. He might be alone in holding that belief, and if alone, then a lunatic. But the thought of being a lunatic did not greatly trouble him: the horror was that he might also be wrong.” -George Orwell, Nineteen Eighty-four*

Let me be clear about a few things up front before telling you how this seems to have happened and what’s going on with this ridiculous moment of academic history—for academic and academic-only it is. This, tedious and difficult to understand as it is, turns out to be centrally important to the present moment and this ridiculous episode, which is but a starkly clear microcosm of a far broader phenomenon by which academic disciplines are being colonized and conquered from within. I contend that this phenomenon represents a potentially existential risk to advanced modern civilizations, and, by the same actors insisting that two and two don’t necessarily make four, am being mocked for saying so. I now hope to convince you otherwise through several thousands of words no one ever should have had to write.

To elaborate, while the most popular assertion being made to counter “2+2=4” happens to be “2+2 can equal 5,” this coming from people including from self-described mathematicians and genuine math educators, among others, the Critical Social Justice activists’ point isn’t that 2+2=5 any more than it is that two and two represents *any* particular quantity. Their point is that 2+2 *can equal* 5, *though it doesn’t have to*. That is, their point is that the objectively true statement “2+2=4” can be deconstructed by means of claiming that it is possible that, in fact, other things can occur too. This allows them to sidestep accusations like that they’re denying that “2+2=4” even while they do it, and (we have to admit) fairly enough because their whole point has *literally nothing* to do with what two and two add to equal.

The activists’ point comes in three stages. First, it is that a statement like “2+2=4” is just one mathematical truth among many, and this seems to be a point that *many* mathematicians who should know far better are eager to help them make. Second, it is that “hegemonic narratives” don’t get to decide it *objectively*, and thus that nobody can say that “2+2=4” is objectively true, which is, of course, patently ridiculous. Third, it is that narratives that have been considered hegemonic in the past or present (e.g., “2+2=4”) should be regarded with extreme suspicion going forward into the future, and people who can make a claim to being oppressed by “hegemonic narratives” *at all* get to have the say on how we should think about those narratives and their specific contents, including simple matters of quantity. That is, the activists are seeking a radical rewriting of the entire rational project, and any reason that doesn’t forward their favored actors as the sole arbiters of what is true and correct needs to be deconstructed by rhetorical tricks and marginalized by moral and, perhaps, physical force and intimidation. They’re seeking a revolution.

This is meant to be accomplished via a distinctly postmodern approach that deliberately removes any sense of stable meaning to anything. In few examples could it be more stark than in the effort to argue that two and two aren’t *necessarily* four that the objective of the postmodernism at the heart of the present Critical Social Justice (or “Woke”) movement is to destabilize any sense of solidity and meaning and then to use the ensuing confusion to advance a particular form of radical politics.

Why is it so clear here? There’s no other reason to deny something so fundamental as “2+2=4” than to generate precisely this kind of confusion, and then into that confusion it is repeatedly asserted that “objectivity” in mathematics, even elementary arithmetic, is the kind of illusion that the powerful delude themselves and others into believing so that they can exclude other possibilities. This statement, of course, divorced from the specific context of what two and two add to equal is a remarkable political tool that could justify literally any double standard or abuse. The name for this approach to manipulating meaning is “deconstruction,” hence my use of this specific term so far, and as it arises explicitly from the poststructuralist ramblings of Jacques Derrida, its postmodern roots cannot reasonably be denied.

### How Did This Happen?

*“He picked up the children’s history book and looked at the portrait of Big Brother which formed its frontispiece. The hypnotic eyes gazed into his own. It was as though some huge force were pressing down upon you – something that penetrated inside your skull, battering against your brain, frightening you out of your beliefs, persuading you, almost, to deny the evidence of your senses.” -George Orwell, Nineteen Eighty-four*

As strange as this turn of events is, the whole affair demands a proper account of how it came to be. The story actually starts in a private text dialogue with someone who was asking me specifically about how postmodernism thinks about objective claims about the world. She asked, at some point, what postmodernists would say about “2+2=4,” specifically asking me if postmodernists would say “2+2=5.” The answer, I told her, is no. The answer is “2+2=doesn’t matter, so long as what it equals isn’t constrained by hegemonic discourses.”

In some sense, the postmodern understanding is “2+2 can equal whatever people want it to equal, and we should be very skeptical of the idea that it equals 4 because so much political dominance is already built into that answer and how it is obtained.” To paraphrase a key point of Michel Foucault, the postmodernist avatar, whether or not a truth claim is actually true or false misses the point that a political process leads to making that determination. For the postmodernists, and their ideological descendants, it is *only* being radically skeptical of this political process that is of relevance, thus arriving at the formulation I gave. This is, of course, what the activists in the present case are doing, being *radically skeptical* of the alleged “politics” of mathematics when the whole program is viewed as a “cultural process.”

This particular radical effort, incidentally, is taken further by the new, more critical (as in, based in Critical Theory) ideology that has adopted postmodern tools, which would take the additional step of classifying a “hegemonic” solution as being indicative of some underlying systemic oppression, particularly exclusion of “other ways of knowing” (like “lived experience”) and “other knowledges” that might say otherwise. That is, in the conceptual operating system underlying Critical Social Justice (i.e., Woke) thought, 2+2 might sometimes equal 4, but we have to understand that accepting this as an objective statement of basic arithmetic contributes to a system of oppression that, in other corners of its existence, oppresses racial, gender, and sexual minorities, women, the overweight, the disabled, and people outside of the “Western context,” which is accused of accepting statements like “2+2=4” in an “uncritical” way (which means without using the favored Critical Theory of the relevant moment).

Pause to breathe. The activists behind this really think like this, and one of the weirder battles of the culture war of the day rages around that fact.

Anyway, to get back to the story, I proceeded to take this thought from my messages to the public in the form of a “Woke Mini,” a line of satirical quips roughly imitating dictionary entries with the goal of exposing and highlighting the inanity of the Critical Social Justice worldview. One of them is for the entry “2+2=4,” and it reads:

*“2+2=4: A perspective in white, Western mathematics that marginalizes other possible values.”*

I tweeted that particular card for the first time on June 8 of this year, and, hopefully, you get the joke. It seemed humorous enough and made my point, so I was content with it, as were many of my followers. What I underestimated, however, was the fact that in cutting far too close to the bone, I had inadvertently introduced a conceptual virus into the Woke Matrix. What happened next is what led us to the present moment in the course of human history.

As it happens, it appears someone put this Woke Mini into the employ of satirically replying to Nikole Hannah-Jones on the fifth of July in response to her tweeting, “I wonder if folks always talking about ‘standards’ ever stop to consider that it’s their so-called standards that are the actual problem.” Hannah-Jones decided to make fun of me by quote-retweeting this delightful troll, including the image of the “2+2=4” Woke Mini, and adding the comment, “Using Arabic numerals to try to make a point about white, Western superiority is just so damn classic.”

For those who don’t realize it, Nikole Hannah-Jones is the architect behind the New York Times’ Pulitzer Prize winning critical historiography called the “1619 Project,” so she’s no small potatoes. It got some attention, not least from people who seem to have dedicated much of their spare time to hating me on the internet in a semi-professional capacity.

In fact, as many people who follow me regularly will know, I have a veritable cottage industry of petty, envious academics who follow me around on Twitter specifically to hate me and to try to discredit everything I do. So far as I can tell, it’s their only hobby, and they’re genuine enthusiasts. One of these, Michael J. Barany, of the Science Technology and Innovation Studies department (read: critical science studies) of the University of Edinburgh, who had previously tweeted (in January of 2019) “1+1=2 is a hegemonic discourse and don’t let anyone tell you otherwise,” replied to “enthusiastically co-sign” to Hannah-Jones’ attempted takedown of my too-on-the-nose satire. (You’ll notice that Barany couldn’t *disagree with or refute* my claim in the Woke Mini, as he had made the same claim himself some 18 months prior and drew attention to it, so he could only sign on to a *problematization* of the fact that I would dare expose this fact about the critical-theory mindset.)

Soon after, also on July 5, “teacher, scholar, social justice change agent” and Ph.D. student Brittany Marshall joined in, apparently not understanding that she was making my point for me. She insisted, “Nope, the idea of 2+2 equaling 4 is cultural and because of western imperialism/colonization, we think of it as the only way of knowing.” This, if you don’t know, is the actual Critical Social Justice view of a “hegemonic discourse” like standard mathematics (including elementary arithmetic). You’ll notice that it’s significantly different than the idea currently being rabidly defended by well-meaning obfuscators on social media and now beyond, who have unwittingly (we hope) adopted the roles of useful idiots, that mathematics just admits a wider range of ways of approaching questions than the basic axioms of number theory. The point really is to create a complete Critical Social Justice revolution in mathematics and mathematics education by undermining any stable sense of reason or meaning. As you can read from a leading scholar-activist in this endeavor, Rochelle Gutiérrez,

Much of what currently counts as scholarship in mathematics education assumes we will work within the given system or expand what we currently count as the status quo. Within mathematics education, we have convinced ourselves that “equity” is a strong enough agenda when maybe revolution should be the goal.

Who goes on to observe that the co-optation of useful idiots in the form of real mathematicians is a necessary part of her revolutionary project:

One thing that has been underscored from this attack is that we cannot create a revolution by ourselves; we need accomplices (not allies) in this work. That is, we need people who are willing to stand with us, around us, so that those who attack us will need to go through them (first). Having accomplices is different than having allies who support with solidarity, cheer loudly from the sidelines, or who safely stand on the sidewalk with their signs. Accomplices do what Delores Huerta called for when organizing for the rights of Chicano farmworkers: “Walk the street with us into history. Get off the sidewalk.” Mathematicians are one group who are showing some promise in the arena of being our accomplices.

She then quotes the following passage about what “accomplices” are expected to do before explaining a litany of ways that even by 2017 mathematicians had become “accomplices” in advancing this revolutionary work: *betray* their institutions and, presumably, their fields themselves, by leveraging support for the “liberation” (this means Critical Theory, specifically neo-Marxism, *by definition*, by the way) effort.

An accomplice as academic would seek ways to leverage resources and material support and/or betray their institution to further liberation struggles. An intellectual accomplice would strategize with, not for, and not be afraid to pick up a hammer. (Indigenous Action, 2014, p. 5)

Speaking of picking up a hammer to take to mathematics, and as a notable interlude in this already weird “2+2” affair, on July 10, an ethnic mathematics teacher and councilperson for the Washington state Ethnic Studies program, Shraddha Shirude, jumped in on the discussion after discovering someone saying something to the effect of adding two apples plus two oranges equals four pieces of fruit—as though the need to group by like units is at all mysterious—is an eye-opening way to challenge people who are rightly criticizing her ethnomathematics (yeah, that’s a word now) program. She wrote explicitly, “This is one of my favorite things to happen upon. Help me respond to all those haters who said my ethnic studies framework claimed 2+2=5… how can we turn this into a true statement?” And, so, the project to make “2+2=5” into a “true statement” began in earnest.

One would hope that the whole thing would have puffed up, been funny and terrifying for a week, and died from there, but such people underestimate two things: first, the fact that the Woke really do think this way and hate for normal people to be able to see it clearly, even though they say it themselves constantly, and second, because I am helping people see their game clearly for what it is, that I must be wrong, problematic, and, above all, discredited completely at all costs. Thus began a relentless attempt by a band of petty Crits to discredit me that has included a professional and academic defense of “2+2=5” that has been raging through the entire month of July 2020. Now, at the time of writing in early August, it seems to have begun to make the leap into the mainstream, necessitating this silly explanation.

### What Are They Saying?

*“In the end the Party would announce that two and two made five, and you would have to believe it. It was inevitable that they should make that claim sooner or later: the logic of their position demanded it. Not merely the validity of experience, but the very existence of external reality was tacitly denied by their philosophy.” -George Orwell, Nineteen Eighty-four*

Although it’s generally true that no one wants to have to stomach more mathematics than they absolutely have to, looking at some of the actual ways these activists and their water-carriers have tried to defend that two and two aren’t necessarily four is obligatory so that the general themes of their activism can be exposed and explained. That theme is this: there is, in every case, some play on words or meaning in at least one of the basic concepts on the table: “two,” “five” or “four,” “plus,” and “equals.” That is, they’re playing word games with the pretense to being profound all with the underlying (and amusingly backfiring) motivation of getting more people to be willing to destabilize meaning and accept deconstruction so that they, the enlightened deconstructionists, can tell people what is right and wrong to think in any given circumstance. Oh, that and to make sure I’m wholly discredited, which is, as I’ve observed, basically some people’s favorite hobbyhorse.

They’ve made an argument to inconsistent units. To paraphrase: *You might say that two apples and two apples is four apples, but at the same time two apples plus two oranges equals four pieces of fruit shows that addition is contextual, and what constitutes the more universal unit is a socially constructed decision, and thus 2+2=4 is not universally true because sometimes it’s 2+2 (two and two) = 2+2 (two and two).*

Explanation: This is more than a bit silly. In applied mathematics, there is no addition possible across unlike units. You cannot add an apple to an orange without reclassifying them both in the same terms, like as fruit. This is important in physical applications, say like describing the acceleration of an object, because acceleration only means anything in terms of how the rate of change of position changes in time (i.e., say, how does the speed in {meters per second} change per second). You can’t directly add an acceleration (with those units) to a velocity (with other units) because it *doesn’t mean anything*. Thus, addition requires similarity of units.

In pure mathematics, numbers are unitless, but the underlying assumption to that is that in being unitless, numbers have *the same* units, and if put to application, that condition must hold. You can only add “like” things together by the definition of “plus.” This seems to score a point for the “contextual” argument, except that it’s just playing a word game. Whether you call the apples and oranges “piece of fruit,” “generic biomatter,” “trash,” or “objects” is irrelevant, so long as they can both be described within that broader category, and the total quantity of whatever category you want to consider is still four, no matter how you slice or dice it. This is a willful application of a little trick, a word game. The point of the word game is to make that seem more mysterious than it is (not at all) for people who aren’t particularly mathematically savvy enough to understand that they’re being had.

They play this same word game in reverse too, intentionally, by employing a shift of numerical bases. To paraphrase: *2+2=4 in base-10 arithemetic, but in base-3 arithmetic, 2+2=11, not 4*.

Explanation: This one is almost infuriating, and it’s a willful exploitation that most people take for granted that we’re working in the “common base” (ten) because that’s literally the point of a common base: everyone gets to take it for granted unless directly specified otherwise. Yet again, this is a stupid word game that they’re using to trick people.

To understand how it gets played, take note that it is no huge mystery to anyone with better than a first-grade education that “2+2=3+1” is mathematically true because both values are, in fact, still four, and four is four is four is four no matter how you write it down. This particular way to write four, “11 (in base 3),” means “3+1.” It is a form of shorthand for the way base-3 (ternary) arithmetic would write the *value* four. In plain English, which mathematical symbols represent, four is expressed this way in ternary: “a single three plus a single one equals four.” This isn’t mysterious at all. The mathematical expression “3+1” is, in fact, one 3 added to one 1, which can be rendered in base-3 (ternary) numbers as (2+2)_3=11_3 (one notation for indicating numerical bases is this, with the underscore or a subscript followed by the specified base).

Activists pulling this wool over your eyes conveniently don’t write the base explicitly and clearly. Instead, they write, “in base-3 numbers, 2+2=11, not 4.” The trick is that 11_3 (one, one, in base-3) *means* 4 as it is expressed in every numerical base above 4. It, 11_3, literally means 3+1 in exactly the same way that 253, in the common base (10), *means* two 100s plus five 10s plus 3 ones (mathematically: 2(100)+5(10)+3(1)). This is a shift in notation of exactly the same type as how “four” in English, “*cuatro*” in Spanish, and “*sì*” (四) in Mandarin Chinese all mean the same thing. Saying “in base-3, 2+2=11, not 4” is a pun on the symbols “11_3” (one, one, in base-3) and “4.” It’s just a word game, not a change in values, and it’s no more sophisticated than if they said 2+2 equals *cuatro*, not *four*, in Bolivia. This doesn’t disrupt objectivity in math, just in the notation one might choose to write it down.

They play this game by shifting to modular (or clock) arithmetic. To paraphrase: *Imagine a clock with values 0, 1, and 2. If you start at 2 and then add 2 more, you go from 2 to 0, then to 1, so 2+2=1 on a clock with numbers 0, 1, and 2.*

This, in abstract algebra (which I had never dreamed I’d have to write a popular essay about) is called “modular arithmetic,” and what the “clock” represents is what are known as “residue classes.” These are the possible remainders when you divide by a given number (in the given example, 3). That is, when you divide any number by 3, you end up with three possible remainders: 0 (divisible by 3), 1, or 2. As it turns out, the usual rules of arithmetic are more or less preserved onto the residue classes, and so you end up with “+” having a natural “modular” analogue when you shift from regular arithmetic to modular arithmetic. For example, 8+5=13, as we all know. If we look, though, 8 leaves a remainder of 2 when divided by 3; 5 also leaves a remainder of 2 when divided by 3; and 13 leaves a remainder of 1 when divided by 3. Thus, 8+5≡2+2≡1 (modulo 3), and 13≡1 (modulo 3), so 8+5≡1 (mod 3), as mathematicians might indicate it. This isn’t a mistake. Modular arithmetic works this way.

So, does 2+2=1 (mod 3) (emphasis on the *equal*)? No. It doesn’t. Both the “+” and the “=” are different in modular arithmetic, as are the meanings of the numbers themselves. That’s the word game. Notice that the binary relation in modular arithmetic is not “=” but instead “≡.” This is because modular arithmetic doesn’t provide *equality* but *equivalence* (in terms of residue/remainder classes when divided by the relevant base value). These aren’t the same “relation” (as they’re called). There’s also a difference in the “+,” as these residue classes, as a set, form a specific group together with *modular addition* as the operation. Modular addition is derived conceptually from the usual addition, rather clearly, but it isn’t the same binary operation and doesn’t even apply in the same mathematical structure. Further, the symbols 2 and 1 here are elements of the residue class of integers modulo 3, not the usual numbers, so they don’t even have their usual meaning. That means these activists are, again, playing a word game. Four, an integer, doesn’t change in value because the symbols “2+2” behave differently in a different mathematical universe.

Mathematicians are content to play fast and loose with the shorthand when the specific context (here, of the relevant group) is clear, but they would also specify that context and accept that without further specification, the common one of usual integers and usual addition, especially if the equals sign is employed, is to be assumed. If wanting to use modular arithmetic, they’d write something like, “in the group ℤ/3ℤ, 2+2≡1 (mod 3),” which indicates to any informed reader that none of the symbols in use are understood in their usual way. It’s an abuse of notation to do otherwise, and any mathematician who is obfuscating on this point knows it because this is junior-level undergraduate abstract algebra, not some way-out-there stuff.

One “former mathematician,” Kareem Carr, of Harvard, took this residue-class argument further, literally to the point of his own absurdity, to establish “2+2=5” explicitly. The thing is, “2+2≡5” isn’t true in *any* modular base except the trivial one, 1. In that trivial base, for the non-maths out there, what’s being asked is to classify all of the numbers by the remainder they leave when divided by 1. That remainder is always 0 because every integer is made out of the relevant sum of ones (that is, by definition, a number like 4 *is* 1+1+1+1). That means that the *only* residue class group in which 2+2≡5 (which still *is not* 2+2=5) is the one in which *the only value is zero*. In other words, in the relevant group, *every number* **≡ ***every other number*, so 2+2≡5 (mod 1) doesn’t tell us *anything* at all. All Carr has managed to do is render an especially meaningless manipulation of symbols to prove a *philosophical* (not mathematical) point that one can play with words to say that sometimes two and two *don’t look like* they add to four.

But wait, you might insist! If these other mathematical groups (or what-have-you) exist, then 2+2 doesn’t have to be 4, and we *do* have other mathematical universes (if we will) in which the Crits are right. No. Again, the symbols “2,” “+,” and “4” *mean something different* in those contexts, and so this is no more profound than misusing a word that has two meanings in two different contexts except that the activists are doing it on purpose, counting on people not to understand it, so they can “expand potentialities of being” and politics, as Foucault would have it. The kindest thing that can be said of this game is that they’re confusing themselves (and others) specifically so they can win an argument on the internet.

From here, the entire “2+2=5 discourse” really started to get silly. Activists desperate to satisfy Shraddha Shirude’s demand to find a way to make “2+2=5” into “a true statement” employed a number of positively stupid tricks and manipulations.

To quote a few examples:

*“There are two factories. Each factory has 2 fully operational machines, as well as half the parts to build another one. If the two factories were joined into one and the two halves of parts were built together, there would be 5 total machines. A case where 2+2=5.*”

No, a case where 2.5+2.5=5, which everyone knows is true for the exact same reason that 2+2=4.

*“Have you ever bought 2 and gotten the 3rd free? Or at least seen such an offer? price of 2 = price of 3. Buy 1 for $2, buy 3 for $4. 1 costs $2, but also 1 costs ~$1.33*”

No, this is a case where the unit price is variable depending on the number purchased but addition is stable given that unit price.

*“Literal-minded people might sometimes say things like I put a rooster and a hen together and I came back a year later and there were three of them (1+1=3) or they might say I left a fox and a hen together and later I came back and there was only one (1+1=1).” *(Incidentally, this one is from Kareem Carr.)

No, here, 1+1 made 2, and then one of the ones either reproduced or was killed, resulting in another plus or minus 1. The arithmetic checks out the same as usual, and adding in this extra confusion about what “literal-minded people” would think doesn’t add clarity, only unnecessary confusion. This, by the way, was given in support of another statement by Carr: *“Statements like ‘2+2=4’ are abstractions. What that means is they’re generalizations of ‘something’. You should always think of these statements as associated with an underlying reality. As a data analyst, I love numbers but it’s my job to connect them to reality.”* (One hopes not this way, frankly.) Do note the explicit intention to redefine what a “reality” is. This is postmodernism, and it’s every bit as ridiculous as usual.

Another example in this vein: *“2+2 doesn’t always equal 4. ‘2’ is a naming convention that aligns extremely accurately with 1000s of years of historical learning and synthesis that knows doubling it almost always equals ‘4’. It’s still not ‘always’ and ‘reality’.*

I’m telling you; they really think this way. They believe that their words and the meanings of the symbols we use to communicate about it create different “realities,” and the “dominant and hegemonic discourses” *unfairly* marginalize and exclude alternatives like this nonsense.

Another “reality” that was presented (by Carr) was that if you are trying to get from point X to point Z through an intermediate point Y, and it’s $200 from X to Y and $200 from Y to Z but $500 from X to Z directly, then because you go from X to Z in both cases for $400 or $500, respectively, two (hundred dollars) plus two (hundred dollars) equals five (hundred dollars). But when you write it down this way, the “reality” falls apart. There are just two ways to get from X to Z, and one is cheaper than the other, and twice two hundred dollars is still four hundred dollars on the cheaper route. The usual arithmetic lets us understand what’s going on.

*“Sometimes, when you put two grams of something with another two grams of something, you get five grams*.” (Arthur Chu)

Nothing in the universe actually works this way, except words.

Several people developed the theme of imprecise measurement, however, in other ways, for example explaining that if your measuring instrument isn’t accurate enough, it might mark 2.3g as 2 and 4.6g as 5, thus, for that instrument, “2+2=5.” Carr himself levied such an example. The problem with this logic is that 2.3+2.3=4.6 is still the relevant, standard arithmetic, and they just need a better scale, a basic understanding of significant digits, and a little more general intelligence or personal integrity (or both).

Things got far worse than this too. To give you a sense of how profound the word games in this activist deconstruction project can be, consider this cleverly idiotic example:

*“What if you wrote it as two+two=five [written vertically] [with] t=5, w=2, o=3, f=1, i=0, v=4, e=6. 523+523=1046”*

For what it’s worth, the stunning example just above was sent to Shraddha Shirude in reply to her request to find a way to make “2+2=5” into a “true statement,” and she, a leader in Washington state’s Ethnic Studies education program, replied, “I love this.”

And finally, my favorite (so far),

*“2+2=5 when the symbol ‘2’ refers to the value ‘2.5’, or more generally, when it refers to something that is equivalent to whatever ‘5’ refers to when operated on by ‘+’”*

This last one one gives away the other half of the activist game (the word game part) so explicitly that it’s literally perfect. It’s a direct explanation that if you just change the meanings of the relevant symbols, you can use them to write something down with a different meaning, i.e., codes exist, and they seem to believe that *everything* is just a code, either for the right politics or the wrong. And this is what they do with everything else in our language too: “racism,” “anti-racism,” “black lives matter,” “rape culture,” “misogyny,” “critical,” “hate speech,” “diversity,” “inclusion,” “fascism,” and “anti-fascism,” being examples that might stand out particularly (though my favorites are “authenticity” and “engagement”). All it says is this utterly pointless triviality: If you change that which a symbol or sound signifies, it means something different. What a shock, then, how often these activists accuse everyone around them of talking in “coded language,” huh?

### Objectivity in Mathematics

*“The heresy of heresies was common sense. And what was terrifying was not that they would kill you for thinking otherwise, but that they might be right. For, after all, how do we know that two and two make four? Or that the force of gravity works? Or that the past is unchangeable? If both the past and the external world exist only in the mind, and if the mind itself is controllable—what then?” -George Orwell, Nineteen Eighty-four*

This has, of course, been very tedious, and maybe few will have read this far, but it’s worth wrapping up with an attempt to defend the objectivity of mathematics and make sense of the tiny little trivial crack through which Critical Social Justice postmodernism is able to wedge all this nonsense.

Mathematics, in its greatest generality, is actually hard to define except in saying that it would be that branch of abstract inquiry that has somehow at its foundations axioms of numeracy. You might think that geometry, in dealing with shapes and forms, offers a counterexample to this, or if you’re more sophisticated that either topology or set theory does, but this, I think, is incorrect. Geometry gives away the game in its name: *geo*–*metry*, the measure of the Earth, and those measurements are relevant to what the field does, even if it doesn’t do it all the time. Topology, even if it were to use no numbers at all, nevertheless derives from explorations about the behaviors of objects called “open sets” first on the real number line or multidimensional real (or complex) space. So it goes with set theory as well, even if no set contains any numbers at all (indeed, one of the key concepts in set theory is “cardinality,” which is an abstract extension of the answer to the question “how many?”). Its usual fundamental axioms, the Zermelo-Fraenkel axioms, were a reformulation of the much more basic Peano axioms of earlier number theory, and other formulations extended from there.

Mathematics therefore derives from axioms, explored logically, and those axioms can either be explicit constructions (as, say, with the axioms of infinity or choice, which are suggested by abstract extensions of numbers themselves) or what seem to be self-evident truths about the world (such as, “there exists a number 0, the additive identity, such that 0+0=0”—if you have nothing, and you add nothing to it, you still have nothing). It is accurate to say that if we change the underlying axioms, or even the underlying logic that operates upon them, we would result in a different mathematics, and these things have been explored mathematically and philosophically, whether they correspond to the real world or not. (A simple analogy is to what the Tufts philosopher Dan Dennett called “chmess,” which are games based on chess but with some changes to the usual rules—games that many mathematicians and philosophers often like to play with, to produce statements of “no abiding significance,” as Dennett puts it.)

Mathematics is in that sense a form of philosophical inquiry where the underlying axioms, thus fundamental premises, are relatively uncontroversial for philosophical exploration. Usually, in fact, the core or base-level axioms (from which others are suggested) are also relatively simple and connect to the real world in a very obvious way, or did at one time and have been extended from there. For example, the numbers, so the fundamental axioms of set theory came from a reformulation of those of number theory, and those are very easily tied to real-world counting experiments. And experiments they are, at least at first: the basic operations like addition, subtraction, and multiplication can be verified in small-value cases empirically with literally no variation ever. Three rocks put together with seven rocks *always* results in ten rocks. This results in being able to define abstractions (axioms) about numerical and related subjects with very little ambiguity, and the mystery of why the resulting systems of truths describe the real world so well is only mysterious because we forgot where we started: with very simple, very obvious observations that admit absolutely no variation from one experiment to the next.

That is, the usual mathematics tends to start with genuinely self-evident truths (we call this “mathematical realism” although it can be done, as Bertrand Russell did at length, through “mathematical formalism” as well), and if we chose to start with different fundamental assumptions, we’d have a different mathematics that doesn’t seem remotely interested in reality at all (these are often explorations within “mathematical nominalism” that divorce mathematical objects from reality, and formalism can feed into this as well). The Critical Social Justice activists are uncritically appealing to mathematical nominalism (and subjectivism) without ever making a case for it, as though its mere existence as school of philosophical thought justifies its status as a destroyer of mathematical objectivity and the entire realist (and other schools) of mathematical philosophy. It’s hard to label this behavior with any friendly terms, so I’ll refrain, but I will say it seems to work on mathematicians who don’t know the full context of what’s going on in this debate.

Though it’s an aside, for an explanation of how this confusion could possibly be widespread enough to turn working mathematicians into Rochelle Gutiérrez’s “accomplices,” one needs to understand that most working mathematicians aren’t all that interested in the underlying philosophy of mathematics. This is for the simple reason that because they mostly have no need to be, as the philosophy of mathematics has very little to do with the actual doing of mathematics (isn’t this always the case?). This fact has been explained, however, by one who is, Ian Stewart. A sufficient explanation appears in his *Letters to a Young Mathematician*, where he wrote, “the working philosophy of most mathematicians is mostly an unexamined Platonist-Formalist hybrid.” There’s rarely a deep exploration, therefore, of what mathematics really is or what it means—realist, formalist, nominalist, objectivist, or subjectivist—on the part of mathematicians, which leaves them ripe for being made into unwitting “accomplices” in an idiots’ revolution. As it turns out, when one’s foundations are largely unexamined, postmodern rhetorical nonsense can pretty easily blow the person off them, and this seems even to be the case with the Fields Medal winner Timothy Gowers, who has unfortunately been dragged into this mess.

Regardless of what system one chooses, once we decide upon our axioms and an operant logic, we can then talk about the “axiomatic system” it generates as “a mathematics,” if we want. Statements within that axiomatic system fall into one of three categories: true, false, or indeterminable. In the usual axioms and logic of basic number theory, “2+2=4, in the integers, with the usual addition” is a provable statement that is *completely true*. “2+2=5, in the integers, with the usual addition” is a provably false statement that is *completely untrue*. “There are infinitely many integers” is an undecidable statement that requires the introduction or rejection of a new axiom to bring into existence (and not all mathematicians accept the axiom of infinity, saying instead things like “the quantity of integers is indefinite”). One could therefore think of the “mathematics” at hand as being the set of provably true statements in that axiomatic system and the project of doing mathematics as finding out what those are, something of a very difficult but very simple sorting question predicated upon truth values, and, however hard proofs, disproofs, or proofs of indeterminability may be in practice, there is *no ambiguity* in the status of any statement in the system whatsoever.

Here’s where the matter of “objectivity” comes into focus. *Within the context* of the axiomatic system at hand, a true statement (called a “theorem”) is completely true and absolutely in no sense false or indeterminable. You’ll notice I did not say it is “objectively” true because I don’t want it to get lost that “objectivity” means something more than merely being absolutely true within a specific context.

The phrase “objectively true” means the relevant statement also says something meaningful and accurate about the world outside of subjective experience—that is, it *depends upon mathematical realism* or at least a quasi-realism through formalism (often branded as “positivism” by the Woke, which they very much dislike). This is precisely what the postmodern effort is meant to break down, though, by rendering it seemingly absurd that we could know anything about the world outside of subjective experience, and it results in lots of double-meaning games played on “objective” happening in the Critical Social Justice analysis, most importantly including the insistence that it means “outside of human bias.” That’s *not* what “objective” in the relevant sense means, however.

Objectivity, in the relevant sense, means *corresponding faithfully to reality*, and thus mathematical statements can be considered *objectively true* to the degree that they faithfully represent reality. The basis of the postmodern question about objectivity is, *is it possible to determine faithful correspondence to reality given that we’re imperfect instruments who necessarily are caught up in our own subjective experience?* The postmodern reply to this question is an emphatic “no,” although the emphasis of that denial is, our own political biases make it impossible to trust any claim on objectivity, so it is those political biases that are of interest. That is, postmodernism is, in some sense, the complete politicization of everything knowable, including the claims to knowledge themselves and the epistemologies used to make those claims. The fact that they can send messages to this effect over the internet, however, is an effective, if not philosophically complete, practical refutation of their philosophy.

Speaking as a realist, reality doesn’t care about any of this immensely human confusion, of course, and so absolutely nothing is changed about basic facts like that—no matter what else—if I am called upon to lift up four rocks two at a time, I will end up doing the lifting action exactly two times to complete the task, leaving no remainder. Further, if I push together two pairs of cherry tomatoes, the result will be four cherry tomatoes, not five, *every time* (no matter how you slice them). This *unambiguously* means that four is twice two, or two twos put together in succession, which can be written down in mathematical shorthand as “2+2=4” or “(2+2)_3=11_3,” if we have cause to use ternary numbers instead of the standard decimal digits. That shorthand carries all sorts of information about what is meant by the symbols involved, but the meanings of those symbols are not ambiguous in their common expression and should be clearly articulated anytime it is not (as with switching to ternary).

A mathematical statement of elementary arithmetic like “2+2=4” is therefore, in fact, *objectively true* (even the water-carrying accomplices agree). The relevant meanings of the terms (“2,” “+,” “4,” and “=”) are stable and unambiguous to the precise degree that it is reasonable to accept that common, and not alternative, meanings can be assumed unless otherwise stated. This, though, is the very definition of “reasonable,” which is the precise definition postmodern epistemology seeks to undermine as being *prima facie* absurd.

Of course, other statements using these or similar symbols with different meanings do not represent a deviation from this objective truth and *may* provide another truth (e.g., that on a rotating “clock” with values 0, 1, and 2, adding 2 turns of the hand twice takes you around once and back to 1). They also may not (e.g., that one rooster plus one hen plus one successfully hatched chick somehow is the same thing as 1+1=3). What determines the difference is also not arbitrary, “contextual,” or cultural but rather the same simple qualification: the alternative is *objectively true* if, and only if, that statement also corresponds faithfully to reality (“faithfully” fails in the chicken example).

### Don’t Be an Accomplice

*“But no! His courage seemed suddenly to stiffen of its own accord. The face of O’Brien, not called up by any obvious association, had floated into his mind. He knew, with more certainty than before, that O’Brien was on his side. He was writing the diary for O’Brien – to O’Brien: it was like an interminable letter which no one would ever read, but which was addressed to a particular person and took its colour from that fact.*

“*The Party told you to reject the evidence of your eyes and ears. It was their final, most essential command. His heart sank as he thought of the enormous power arrayed against him, the ease with which any Party intellectual would overthrow him in debate, the subtle arguments which he would not be able to understand, much less answer. And yet he was in the right! They were wrong and he was right. The obvious, the silly, and the true had got to be defended. Truisms are true, hold on to that! The solid world exists, its laws do not change. Stones are hard, water is wet, objects unsupported fall towards the earth’s centre.” -George Orwell, Nineteen Eighty-four*

The common meanings of mathematical symbols like “2,” “+,” “4,” and “=” are also not arbitrarily chosen. They have arisen not out of the abstract ether or any particular cultural context but from the bare fact of reality that once we define a quantifier “1” to represent the quantity of a single object of a given (perhaps broad) description, 1 and 1 make 2 and 2 and 2 make 4, *by definition*. “Four,” as a quantity, no matter how it is represented, literally means 1+1+1+1 in that system of thought, and that system corresponds to reality perfectly faithfully. The only “cultural” processes involved on any level are which cultures happened to be first (Indians and, by way of India, Golden Age Arabian Muslims) to use particular grunts and squiggles to mean the things that we take to be the values two and four, the binary operation plus, and the binary relation equals, and which cultures currently use them (basically all of them). That has no impact whatsoever on the basic fact that all of these have clear, unambiguous meanings that are ultimately derived from simple observations of facts about reality that are true in every conceivable cultural setting (at least in this universe).

This means that “2+2=4,” once the symbols being used are explained in the context they are intended, is *objectively true* and *cannot be false*. By the same token, “2+2=5” is *objectively false and cannot be true*. The only way in which this can be changed is by changing the underlying definitions of the symbols themselves, which means by engaging in a fallacy of equivocation that could be resolved—rather, exposed—merely by offering full transparency on what each symbol means. (This might remind the reader of “trans women are trans women,” which, despite being a tautology, is a higher resolution statement than “trans women are women,” which is a word game being played disingenuously with the word “woman,” which represents a particular kind of category that can be defined, clearly, in more than one way.) By means only of this mere equivocation, mathematicians and others are being made into “accomplices” in a revolution they, I would dare guarantee, would not otherwise support.

So, now I’ve written well over 8000 words on the stupidest topic I could possibly have imagined ever having to write about, but it matters—and there’s a point to take away from all this. It is that postmodernism, particularly in the hands of the ideology of Critical Social Justice, *is not at all interested in truth*. It is only interested in power, which it will establish through its attempted revolution, which it in turn knows it can only achieve by turning otherwise intelligent, well-meaning people into “accomplices” by manipulating their good will, charity, fear of being disliked or ostracized, and, especially, unawareness of what is actually going on beneath the rhetorical tricks they’re being served up with intentionally limited context.

To achieve this revolution, postmodernist Critical Social justice is centrally interested in destabilizing meaning so that those it anoints as sufficiently virtuous can decide what is “true” and so that no one has any grounds upon which they could disagree, even in principle. It is a direct assault on reason itself by means of destabilizing the meaning of *meaning* with the purpose of installing its own priestly caste of arbiters of how things will be according to the rubrics it lays out. This is, however, as I claimed at the beginning, a breakdown of the fundamental logic of civilization, which depends entirely on the ability for each citizen to generally understand something of how that civilization operates. It is also a replacement of that fundamental logic of civilization with the fundamental logic of something more basic and less able to meet the needs of the people who will still be forced to live within it: self-interest, cronyism, corruption, and an unstable form of uncivilized might-makes-right that will surely eventually collapse into the more brutal and familiar stable sort in which whomever can kill enough people gets to make the rules.

*“Freedom is the freedom to say that 2+2=4. If that is granted, all else follows” -George Orwell, Nineteen Eighty-four*

Maybe don’t be an accomplice, then.

## 129 comments

“Two and two make four,

“They never make five,

“And as long as we know it,

“We all can survive.”

From “Man in the Mirror,” from the 1970 LP “Songs for Beginners” by Graham Nash

Makes me homesick for the hippie far left of the past.

Yes – and a true piece of experiential word craft like that is a proof of the days leftist reality also = the left in those days expected 2 + 2 to equal FOUR I.E there was no correlation between todays left with thier silly 2 + 2 is 5 essentialism // and the hippie left circa late 60’s.

HOWEVER – James Lindsay has not only been insisting theres is a direct correlation & identical ideology ( as has Lancing ) Lindsay has also been regurgitating CIA propaganda to back up the notion.The CIA must be delighted that a load of crap they wrote up in the late sixties has become the anti woke bible thats actually THE WOKE BIBLE!! The only problem is whether the CIA wrote a masterwoke THAT INCREDIBLE – or is it that lindsays stupidity is that incredible?

I ask only since as far as i’m concerned the ‘left’ ( of the 60’s ) behaved and spoke remarkably like those accused of being far right today. Marcuse had not influenced college kids according to the CIA docs that have been sourced as material by this site. On the contrary, those docs were developed specifically to fool hence socially engineer the public into the belief that such had taken place when it had not. Marcuse in fact was WARNING students not to listen to such propaganda, and that is why he was screwed over. Marcuse is a trouble maker and was left field ( ofc ) -but he wasn’t that kind of trouble maker. In fact he had the courage to speak against ideas that were to his general advantage should literal anarchic desire be it.

So yes i agree with you about the way the left in those days were generally pragmatic and peaceful if young & silly like most of us are. I find it completely ridiculous that a so called ‘anti woke; site ( like this one ) is in support of woke generating // creating // booting up // catalytic // CIA lies

The reflexive property in mathematics means that every element is equal to itself. For any given element \( a \), the equality \( a = a \) holds true. This property is fundamental in mathematical reasoning and proofs.

I know the woke don’t use logic but I was thinking about the chicken example of 2 + 2=5. This concept validates the life of unborn children. 1 + 1 = 3 right?

I cut a piece of wood 2ft long then cut another 2ft long from the same piece of wood that is 4in wide. I then put both pieces of wood end to end and for some reason I still keep getting 4ft long. I’ll keep running this experimental process till I find the mythical 5ft. Wish me luck.

The value of 2 is feet and the value of 2 is door combined, equaling the value of 5 which is get away from me you are a waste of time and space…

Math makes my head hurt, so I use a calculator. The calculator says that 2+2=4, therefore, that’s the answer.

Unless you are grotesquely unable to visualize that which is real 2+2 will always equal four

No one mentions that these discussions are based upon work within the decimal system. The decimal system is, in fact, a system. It has its own rules. If you reject the rules, then you reject the decimal rule book. To do so for any reason outside the system is a non sequitur and not worth the time to explain to anyone, either the righteous decimal folks or the subversive operatives.

It is tiring how smart people are drawn into arguments with provocateurs. If you engage with subversion you are subverted. It is more tiring listening to the director preach to the choir. I guess it sells books, but what is the intellectual value of pretending the subversives actually need to be explained, as though they are stupid and we smart people can “virtue signal” our smugness. If you stray from the core logic of a matter by entertaining crazy talk, you lose the discussion.

If this is writing and analysis by someone claiming to be an author and mathematician I can definitely successfully enter the industry with such a low bar…regardless and independent of what the woke crowd tries to achieve have you ever heard of “logical positivism”, falsifications principle proposed by Karl Popper…the demarcation problem?

What you are doing here is scientism at most

That person makes no factual statement with which we could argue, but only an ad hominen attack. This is a textbook attack of the collectivists. By the way modular arithmetic is fun in particular with finite groups and primitive roots (how with two numbers you create a whole set of numbers. That’s fun).

Well….

I propose the following: whenever a Woke orders an apple juice for him /her/they , and another one for his/her/their friend, supposing one apple juice costs 3$, ask them 9$ for their two apple juices. You’ll see how they count!

You are an activist against left leaning activists. You are no better (and your math is wrong as pointed out by others lol)

What part? you may have had difficulty with that 0 + 0 = 0? I know it’s awful. Let’s try 4+0 = 4 and 4 + 0 – 4 = 0, in which Zero is the neutral/identity element for addition, and don’t get me started with the multiplication. That rule is part of algebraic structure. Number can be written the way you want but not their rules.

I live in Taiwan, and over here in Asia, 2+2=4. There is absolutely no debate about this. Perhaps this highlights why the Asians and specifically the Chinese are eating our lunch. They don’t waste time on useless pseudo-reality.

The article uses the equation “2.5+2.5=5”. This is sloppy. What is intended is 2.5+2.5=5.0 . By missing the “.0” the equation is equating the sum of 2 real numbers, which have tolerances, to an integer, which doesn’t. Does this matter? Yes. The arguments around 2+2=5 are all based on imprecisely applying the rules of mathematics to try to subvert the reality that notation is a social construct, but the underlying maths it represents is objective, and the same for any race in any part of the universe.

absolute nonsense. 5 and 5.0 are the same thing and they do not need to be distinguished by a decimal. You’re conflating how integers and floats are typically distinguished in most programming languages. This distinction doesn’t exist in actual mathematics, it is purely an artifact of the need for a computer to know how much memory to allocate for different types. 2.5 + 2.5 = 5 is a perfectly valid mathematical statement.

Integers are a subset of decimal numbers. 5.0 is one of the decimal numbers that is also part of the subset of integer numbers. That’s it.

Absolute nonsense. You should have been paying attention in class when they got to the stuff about error banding, and expressing precision in answers. I was taught to distinguish integers from reals in maths classes, long before I met a computer.

coppice is wrong. The integer 5 is exactly equal to the real 5.0. 2+2 = 5 = 5.0, and 2+2.0 = 5, and so on.

5.0 is somewhere between 4.95 and 5.05. 5.00 is somewhere between 4.995 and 5.005. 5.000 is somewhere between 4.9995 and 5.0005, and so on. So sayeth maths books about the notation we use today.

Not true, writing 2 then lying that 2=2.5 because that is what you meant, is farcical. But prove me wrong by telling your employer to pay you your teacher to grade you 1/5 etc less because that is what they intended etc

Mathematically, you have the right to establish a relation between R (real) and N (natural). You use a subset of R, Q is a subset (that’s how we name them in France). Q = rational numbers. So 5/2 + 5/2 = 5. QED. but we have understood that no mathematical demonstration will satisfy you, because you didn’t try to help him with the article but accuses him of incompetence.

You’re right in the context of a scientific measurement, where expressing significant figures is relevant, and wrong in the context of pure mathematics, where it is not. This is a nitpick at best, and a rather strange one at that. If you’re making that statement in 100% good faith, then this might simply be a case of an “unknown unknown,” where you and the person you’re arguing with are operating from two correct but distinct understandings of how many decimal places need to be included in a sum, based on context.

The mathematical reasoning in this article is based on assumption that there is only one arithmetic. However, mathematics is developing. For millennia, people thought that there was only one geometry. Then non-Euclidean geometries were discovered. Even longer, people thought that there was only one arithmetic. Nevertheless, at the end of the 20th century, non-Diophantine arithmetics were discovered. In some of these arithmetics, 2 + 2 = 5 or 1 + 1 = 3. This this rigorous mathematics.

The context of the article is clearly arithmetic as considered through the CT lens.

a) The other comment is right, this is clearly in context of Diophantine arithmetic.

b) non-Diophantine arithmetic is not somehow a counter-proof on Diophantine arithmetic and is analogous to path-algebras where something like “order of symbol” and domain of operation are specifically accounted for.

The foundations of Non-diophantine arithmetic are still rooted in diophantine arithmetics by the simple fact that we need to make well ordered arrangements of symbols to represent the meaning encoded within the arithmetic.

It doesn’t “undo” 2 + 2 = 4 or mean that it’s another “way of knowing”, it’s simply something completely different that uses a similar set of symbols because it’s tedious to memorise many symbols.

It’s actually something else.

What it does do is contextualise the scale problem of networks/graphs within a field of operation.

In other words, it accounts for how representation of a symbolic logic effects that symbolic logic like call and response in music. We play the room as well as the instrument, ourselves and the audience.

Again, to re-iterate, the “2”, “+” and “=” signs in non-diophantine arithmetic are not the same thing as in diophantine arithmetics, and these kinds of conflations of representation and meaning have been completely covered in the article.

Trying to make unique symbols for every single case is pragmatically pointless.

The symbols are arbitrary. In themselves they don’t mean anything.

Only when you make sense of them do they mean anything, and when you actually sit down and do the maths the painfully obvious inadequacies of your kinds of conflations fall out straight away.

Standard arithmetic orderings are embedded within all alternative algebras because you can’t have an algebra without some sort of ordering under which to establish a relation.

The type/class/category of ordering comes as a consequence of the establishment of the type of relation, and if you don’t establish a relation then blank. We simply can’t do anything.

All the non-diophantine arithmetics do is form relations between partial mappings within the Natural numbers that cannot be defined without some form of diophantine arithmetic.

Whether you were aware of otherwise, your argument was just an obfuscation.

You don’t understand the nature of knowledge and I will explain but first thing first. We know that kind of argument – there was something different before therefore it could exist something different in the future, and therefore again it is vital that we keep an open mind. No we don’t, it’s not how it works.

Knowledge do not replace knowledge, it is integrated into a network in which we link each fact, true, statement, etc… Even if a super theory in mathematics happens, it will have to integrate the actual mathematics, because that mathematics is linked to thousands other knowledge. But yet again, we know that your understanding of what we say is subordinated to your ideology, which is linked to your emotions.

If 2+2=5, then 2=3, and 1=2. Therefore every $1 of salary you pay the SJW is equal to $2. Therefore they can take a salary cut of half without any ill effect whatsoever.

The simplest way to put this to the test is to ask an adherent of this new woke mathematics whether they’re willing to let you lend them $200, then $200 more and ask for them give you $500 back.

All terms up front, naturally.

This whole argument is actually a really good demonstration of the internal contradiction of postmodernism:

If 2+2= anything. Then 2+2= infinity

If everything can be interpreted in an infinite number of ways then that, itself, is a universal. But universals are incompatible with postmodernism. Ride the spiral.

Sorry for not reading everything, i skimmed over most of it, since i saw serious flaws from the beginning. Some other commenters already pointed out the lack of Gödel, Wittgenstein, Einstein (non-Euclidean geometry). I would add Hegel, because of the lack of reflection about the fundamental notions of Quantity and Quality. Other main notions i failed to recognize: synergy, part / whole (“The whole is more than the sum of its parts.”), discrete / continuum

So what you have done is not philosophy. It’s a form of dated, but somewhat acceptable mathematics. Ok, in the logic part you missed modal logic, but that’s the postpostmodern world: Noone knows everything. Though a problem arises, if people act as if they knew everything.

This considered i want to turn to the concrete examples.

The main problem i see in this essay is missing one substantial point of the postmodernists completely: They don’t question the mathematics as a logical corpus per se, but the naive, quantitative and f.e. reductionist and therefore problematic application onto reality. The abstraction distorts the understanding of concrete reality.

1. “2 factories + 2 factories = 5 factories”

This is a problem of abstraction from hidden properties in the underlying qualities. The process of quantifying tries to count the functional entities, therefore the broken machines are not counted at all (= 0), and become “hidden properties”. So this cannot be quantified naively with 2.5 + 2.5 = 5. Because a factory is a whole organically composed of different parts. Maybe one machine is missing the engine (the quality counts, but cannot be counted), and the other factory has a spare engine. So only in such a case put together they can provide an additional output (+1).

2. “2 + 2 apples” – counting apples

There is always a problem with abstraction, and i only try to show a rather simple example of the literal limits of such an approach.

a) The first case of counting is if it’s just subjective, without actually reflecting about what the whole unity of the sum actually entails, i.e. if i imagine the apples in an abstract space, not interacting with each other. This in itself leads to a wrong perception, it just doesn’t show, as long as you work with small numbers.

b) Because apples as physical entities also interact with each other. They’ve got mass. So adding apples increases their mass. 4 apples are no problem, there is still room in the basket. At some point the basket will break, but that’s not what i am pointing at.

Let’s go to a million apples and fill a crater with it. At some point the mass of the apples on the top will put so much pressure on those on the bottom, that latter are turned into apple puree. And you are no longer able to count them anymore. (Quantity has reached a limit and the Quality of the elementary unit has changed. The apple is no longer an apple.)

There are many more examples and basic flaws of a naive quantification of concrete reality (see list of notions in the beginning).

I am not a postmodernist, quite to the contrary. But even worse is a naive and regressive rationalism, which on the one hand fails to describe reality properly and on the other falls victim to a superiority complex, attempting to teach everybody else, how it’s done.

You’ve missed the tree for the forest.

The point of the article was not to engage in a comprehensive treatment of mathematical philosophy, but to highlight the fallacies of the CT approach to arithmetic. Perhaps you should consider the article less of a speedbump on the way to the comments and spend a little more time in the content?

Regarding your point 1. The problem is attempting to measure machines, not factories, thus assuming definitions about what constitutes a factory are irrelevant. The rest of your argument are maybes and assumptions, which occur naturally due to the nature of how the problem is worded. If there are hidden variables, then the problem – to be useful, and not dishonest – must account for the variables. Since the problem doesn’t do that, the reader is forced to fill in the blanks as you have done. Neither your position nor Lindsay’s is more valid than the other, given the formulation.

As to point 2a, this is sophistry. You’re introducing irrelevant qualitative complexity. For 2b, this is more of the same, but now you’re introducing variables that take us well beyond simple, abstract arithmetic (the point of the aritcle) into calc-based physics.

In short, you’ve taken the problem as presented and repackaged it into what you want to talk about, not what’s in the article.

If 2+2 does not equal 4, then all stats indicating oppression and all stats keeping us shut down for Covid go out the window.

Thank you for this article. Having left my education a long time ago (graduating in 1979), before all of this madness starting to really take hold, I have been trying to come to terms with what is happening around me. Doubt started to appear in the late 90’s when my kids moved out of the home-schooling environment to the public school system. By then several ‘ideas’ had become established in the curriculum, the first is that children were no longer taught about phonics when being taught to read, the accepted practice was to read to the class and somehow the knowledge of how words/sentences were constructed would be transferred by following along with the text. The second was emphasis on getting in touch with their inner feelings during their classes, including a mandatory one on ‘personal developmental relationships’ – ‘how does this make you feel?, so you feel safe or threatened in anyway?’. In hindsight seems to be laying the foundation for the current push for people to accept ‘lived experiences’ as truth and the oft heard lament of lack of safe spaces for students entering college.

So my ‘lived experience’ is nothing like what the SJW’s portray for white, anglo-saxon, males. I have no university degree (by choice) and have worked all my life in a technical trade (O&G related where mathematics played a very important role). I have traveled all over the world and experienced different cultures and societies. I have to look really, really hard to see any form of oppression that is supposedly so prevalent on our society.

I had no knowledge of many of the people who appear to be leading this crusade against Western civilization (Robin D’Angelo, Ibram X. Kendi, etc.). Lately I have been seeing more and more articles about ‘alternate ways of knowledge’ and today is the first time I have made note of the name Rochelle Gutiérrez, so I turned to Google. The first thing that jumped out at me was a quote she has apparently posted online, most of you have probably seen it – “On many levels, mathematics itself operates as Whiteness. Who gets credit for doing and developing mathematics, who is capable in mathematics, and who is seen as part of the mathematical community is generally viewed as White”.

Anyway, the main reason for my response (other than to thank the authors of this site for helping me to understand), is to offer up something I found from the Google search confirming (in my opinion) that ‘useful idiots’ do exist and spreading the damage. Here is a link to an article published by a small Liberal Arts college in Minnesota.

https://thecarletonian.com/2019/10/18/rochelle-gutierrez-shaking-math-education-to-the-core/

Scary stuff. Once again, I say thank you for providing clarity and helping to understand ‘the enemy’.

“Twice two is four, but twice two is five is a charming little thing, too…” Fyodor Dostoevsky, Notes from Underground

And now for something not at all different: Moving Towards a Feminist Epistemology of Mathematics

https://www.jstor.org/stable/3482752?seq=1

I admire your efforts to take down the Woke infrastructure. Perhaps our best ammunition in this war are viral injections of nonsense. This inoculation may help prevent consensus from building inside their own rotten community and further reveal the absurdity to persons of reason.

Not enough. Take it a step further. Imagine you discovered a means of manipulating a giant infrastructure and population of extremist conservatives in liberal clothing. What all could you do with that?

James, you’d make a great presuppositional Christian apologist.

I used to work as a geophysicist in the oil exploration business. Unlike quantum mechanics, geophysics is a very inexact science and when developing a prospect, the powers that be always told us to be optimistic and let them decide whether to spend the money to drill a formation. This whole discussion reminds me of a joke that passed around the exploration departments that went something like this:

Three guys (or in today’s terminology humans) are interviewing for a job, an engineer, a geologist and a geophysicist. The engineer goes first, and at some point in the interview they ask him, “What is 2+2”. The engineer says, “Oh, it’s 4.000000.” Next up is the geologist, and he gets asked the same question. The geologist says “” Ahhhh, it’s somewhere between 3 and 5″. Finally comes the geophysicist and he gets asked the same question. He answers, “Well, what would you like it to be?”

Black Fives Matter

Thank you, Dr. Lindsay. Any opportunity to debunk and denounce postmodernism for the diabolical menace it is (and I use “diabolical” in conscious, deliberate, literal intent here) should be taken.

It’s the hypocrisy that galls me more than anything else. The only way to make “2 + 2 =/= 4” even semi-plausible to enough people to accomplish its actual sociopolitical end is to use a telecommunications and technology network designed on the basis that 2 + 2 = 4. The very medium of their argument contradicts it.

(Which actually suggests a very mischievous way of countering it: write a little code routine that randomly shifts the ASCII value of text characters by one in a random direction every time a message is uploaded, and insist that anyone taking a postmodern approach to STEM has to use it. Make the opposition live up to its own rules, as Alinsky noted.)

Here’s another 2+2= example, from an article about hierarchy & conformism in Japanese society, and the novelty there of the notion of the Western ‘individual’. Apparently shakai (society) and kojin (individual) only entered the language as improvised translations for the English terms shortly after the beginning of the Meiji era (1868–1912).

“What Japan had instead of a modern society was seken, a loose term referring to the general public. And within this Seken were Mura, or “villages,” to which everybody belonged.

[…]

Mura derives from the verb meaning “to flock together” – an action taken by people without the maturity to stand up as individuals. That is to say, a mura is a flock of people with zero identities. The Japanese painter Fujita Tsuguharu (also known as Léonard Foujita) was active in Paris for almost two decades from 1913. When he was driven out of the Japanese art world after World War II, he returned to France and became a French citizen.

In his later years, he wrote: “Even if public opinion [seken] is strident, spreading from one mouth to thousands of ears, I say that however many tens of thousands of zeroes gather, they only equal zero and will always be less than one.””

In your face, Lindsay.

After all that fine logical argumentation about agreed-upon conventions, that next to last sentence of yours made me wince a bit:

“… in which whomever can kill enough people gets to make the rules.”

Whoever, not whomever! Whoever must be in the nominative case, as the subject in the phrase “whoever can kill enough people” (which is itself the subject of “gets to make the rules”).

Otherwise, full marks. 😉

Funny that the true source – the manual – for all the subjectivity and relativism of social justice as a means of pushing a new reality is the visual arts. The methodology seems very similar. The lack of talent, intense envy and hatred for the talented (and their fans) caused many artists and critics to create arguments to destroy all the social functions that once dominated the arts and the teaching of the arts. Where allegory and metaphor was once used artspeak and utter nonsense now dominate. You would think that what is visually set before your eyes is reality, something which could be objectively ascertained and discussed, but what we have today, taught in every university of the West, in the visual arts department ( or Fine Arts) are methods of arguing against art with social functions that benefit the public in favour of arguements that benefit the talentless artists. Now anyone can be an artist. No one is special. Now that millionaires and billionaires support so much art that is subjective nonsense there is little hope that contemporary galleries will support anything else. Obviously not in all cases, but predominantly. Meanwhile popular culture flourishes. And all the perennial functions of the arts can be found there. The talented have moved to Hollywood. And we know how much contemporary artists and so many others in the humanities hate upon Hollywood. I’m an artist and I’m waiting till Covid-19 subsides in order to better consider a move to Los Angeles to better my chances at a more lucrative art career, for here in Canada support for the absurd contemporary art continues unabated. I’d love to hear what you think about the last 150 years of art philosophy and criticism. Pick up any contemporary art magazine and you will wonder how in the world so many intelligent adults can swim in such a pool of nonsense.

Thanks Doc, I know we’re not supposed to have heroes, but you’re it. Keep up the good work. Take care.

The problem with the “You’re changing the meaning of words! Stop doing it!” strategy is that it’s a loser. The woke community changed the meaning of “marriage” and loved the results. In fact, the entire culture came around to their new definition, including many of the people complaining here about “changing the meaning of words.” Same goes for the meaning of “woman.”

Saying “2+2=5 is fake because you’re changing the meaning of words” is exactly analogous to saying “Caitlyn Jenner is a woman is fake because you’re manipulating the meaning of words.” Yet look around. No one wants to be the neanderthal bigot complaining about “changing the meaning” of “marriage” or “woman” because gay marriage and transwomen are wonderful things we all support now. Right? And that seems to imply that changing meanings is perfectly okay — something most of the public agrees with, including most conservatives.

The reality is that word meanings change, so “you’re changing the meanings of words!” is not an effective counterpunch. It’s more like flailing. What’s needed is sociological understanding of *how* the woke army succeeds in its redefinition campaigns. About 75% of it is topdown propaganda, which is why the left is dead serious about maintaining control of the media.

Your points are intriguing, but I’m not convinced.

Marriage is a social institution that has had many forms over time and space. But that is all it is. There is no one true form. The battles over gay marriage were about changing the legal and social reality in Western democracies in the 1980’s and afterward.

Of course the push for the legal and social acceptance of the idea that “transwomen are women” is also about changing the legal and social reality in Western democracies. But there is this pesky notion that in a sexually dimorphic species such as ours women “really are” adult human females. And the contrary notion that has arisen among some members of the human race, with our newly evolved comprehension, inhabiting a truly insignificant speck in this vast universe, “really is” hubris of cosmological proportions.

Things don’t belong to linguistic categories in themselves. Whether a person is a woman or not hinges on the definition of the word “woman” in the pertinent linguistic community, and that definition is not a characteristic of the person concerned. If there are conflicting definitions in play, then there is no objectively “correct” answer to the question “Is X a woman?” Objectivity comes into play only after definitions have been agreed upon.

Asserting that X “really is” Y is just another way to insist on a definition of a word — to portray a favored definition as fact, and thereby gain the persuasiveness of factual status. (By placing “really is ” in scare quotes, you seem to appreciate this.) It won’t work, however, when the opposing side controls all the social machinery of word definition: style guides, media programming decisions, terms of service, expert opinion, academic gatekeeping, hate speech laws, deplatforming/firing power, dictionary committees, etc. If James were to sell T-shirts that say “Caitlyn Jenner is a man” rather than “”2+2=4”, this powerful apparatus of word usage policing would quickly mobilize and shut him down. My point is that you can’t win by insisting on what things “really are”; you actually have to control the apparatus.

The Wokerati’s examples where they allegedly show that 2+2=5 seem to me the equivalent of stating: Replace the symbol “2” with the letters “cous” and you will see that it’s possible that 2+2= Middle Eastern dinner for five. It’s just that dumb.

Kareem Carr’s is an interesting variation on the divide-by-zero trick a professor of mine once used to prove that 2=1 as a joke.

“Genuine utterances about the nothing must always remain unusual. It cannot be made common. It dissolves when it is placed in the cheap acid of mere logical acumen.”

– Martin Heidegger

Sorry, but Heidegger was a Nazi, who wrote Nazi Philosophy and till his end regreted that Nazism didn’t make it, but believed it was the good solution. All Post-modernists and alike in France followed and still are, what is called Heideggerianism. We know what the result of post-modernist looks like today, and it’s not good, not good at all.

I just discovered your blog and it is just what I was looking for: a site with arguments to dismantle all this absurdity that we see today. Sometimes JSCs are so tenacious that they end up confusing.

Thank you very much for your efforts, from Spain.

“If being complicit is wanting to be a force for good and to make a positive impact, then I’m complicit.”

Interesting to see that the Wokes and the Trumps are playing the same game.

Not sure what you’re specifically referencing with the Wokes and Trumps but I find the political spectrum to be more of a circle than a line. Go far enough in one direction and you’ll eventually meet up with the ‘other’ side.

There are a few people out there who must be on the same page with numbers:

– the doctor writing prescriptions

– the pharmacist filling them

– an accountant

– the person who designed the timing system in the car

– the person who answers 911 and does not think I called to order a sandwich

The thing is that this is not just about math and the hegemony of whiteness in the application of numerical value. This is also about the deconstruction of language. If words no longer mean what we think they mean (and this is not just a question of two people misunderstanding each other), it becomes impossible for us to talk, to be neighbours, or to conduct basic business. It will lead directly to the disintegration of society, as misunderstanding each other will be the norm.

Two other thoughts, and then I will go study with my son…

Regarding language, grammar and usage are important to meaning. Consider the following two sentences:

I would like to thank my parents, Dairy Queen, and Burger King.

I would like to thank my parents, Dairy Queen and Burger King.

In the latter sentence, my parents are Dairy Queen and Burger King.

A basic axiom of math should have been stated in this article. It is the reflexive property of mathematics. That property states that in any equation, A = A. Most of the academic dribble noted in this excellent article can stand only if that basic principle is ignored.

Good evening to all.

-RavSean

The poster used to make the point doesn’t say 2+2=5. It says 2+2+”the enthusiasm of the workers”=5. The 2s reference two years and the poster illustrates the idea that the workers could compress a Soviet five-year plan into four. More on this here: https://ru.wikipedia.org/w/index.php?title=……_…_%E2%80%94_….&oldid=108567099

Yes, but you could still argue that this is an example of the Soviet government telling a lie. If the 5-year plan was so good, they wouldn’t need to tell workers to make it happen in 4 years. The government would have made a 4-year plan. By using 2+2=5, they are deflecting criticism away from the government, and adding an implied threat: that if the government says 2+2=5, or that a 5 year plan can now be done in 4, you Soviet citizens had better not question it.

Yes but in the nearly 80 years of the USSR, ALL of their “five year plans” equaled FAILURE and DEATH….. in numbers much much greater than FIVE….

Some of that “accidental” like Lysenkoism, not counting the realist agri scientists who were killed for treason. Lysenko was good at being a suck-up to Stalin and the Party.

Some, intentional, murder and scapegoating to apply terror as an ingredient of production.

Some CIA reports that reformer Khruschev told Mao to stop with the starvation quotas. Mao’s response was to increase death quotas and denounce Khruschev as disloyal to M-L.

Why isn’t Gödel mentioned even once here? Not in the text, nor in the comments?

I’d say that Gödel first, and especially second incompleteness theorem could easily be used by the author’s opposition to make a serious dent in his narrative. Simply put, any logical system powerful enough to produce integers has unprovable truths and un-disprovable falsehoods. Further (and even more relevant), such a system can’t even be assured not to contain contradiction (the canonical one being 0 = 1).

0 = 1. Put that in your 2 + 2 = 5 pipe and smoke it.

Now, I’m simplifying to make a point (then again, so is every statement of this article). My point here is not to give ammunition to the opposition, but to point out that the strongest attack to the author’s point has been left defenseless,

And if anyone is wondering, then, what my point of view is on the issue actually is I’d say that it is that there’s no actual issue at all except the issue of political polarization that brings out the dummest even from the smartest of us.

I had the same reaction. Gödel’s incompletness theorems demonstrated the inherent limitations of any axiomatic system used to model arithmetic. That doesn’t mean that the activists are right (indeed, I don’t think they are), but it does mean that Lindsay should at least acknowledge that the 19th Century model of mathematics as a bastion of purely objective truth is no longer supportable.

The second Godel incompleteness theorem does NOT assert that provable statements do not exist. NOR does it say that any system powerful enough to produce the natural numbers is incomplete. I note that Presburger arithmetic is complete and was proved as such a while back. So, no, that is NOT the strongest argument of the crackpots denying that (2+2) = 4. The Godelian incompleteness theorems in their original formed referenced that a fair amount of arithemtical truths *were* proved in Principia Mathematica or could get proved rigorously using some small changes to the symbolic system.

Also, (2+2) = 4 can get proved in Presburger arithmetic, which again is complete.

The way I understand this Lindsay’s text tells me that he carefully limited his speech to the logical deduction from clear and simple axioms. Gödel’s incompletude works outside the axioms. Your objection does not hold.

I’m not a mathematician. I have a humanities background. So I approach the issue this way. Let’s start with the basis of counting, the number one. The number one derives from our ancestors’ first realisation of the individual person. Therefore, one = me alone, or you alone. Together we become two. Thus counting is embedded in both self-realisation and the ethical realisation of the other. Two takes us out of solipsism. (Anthropologists believe that the first counting went, ‘one’, ‘two’, ‘three’, ‘many’. ) If wokists wish to deconstruct two plus two equals four, by implication they are trying to deconstruct the number one which underlies the number two (one plus one equals two). By so doing they deconstruct not only the number one and the basis of counting for all – not just white – societies, but also the individual human which is the basis of counting. And by deconstructing the number two they remove the discovery of the unique other person who is not me. But perhaps that’s the intention.

Possibly, but it’s also a technique used by liars, con=men and abusers to weaken their targets’ resistance to the con. If you can convince people to repeat a blatant lie, while threatening or scaring them at the same time (often by punishing a lone person who resists the lie), they will convince themselves that the lie is true out of a sense of self-preservation. If you convince the majority of people to repeat the lie, it keep naysayers quiet and compliant, freeing up the abusive liar to get away with their nefarious activities. Also, if you can normalize this kind of cognitive dissonance in people, you can get them to keep doing it when you try to take advantage of them the next time. It’s a means of abuse and control.

Another rigorous system with irregular “truths” like 2+2=5:

Form the sequence F: 1, 0, 3, 2, 5, 4, 7, 6, … (just reverse the order of adjacent pairs in the ordinary sequence 0, 1, 2, 3, …)

Let 1 serve as “0”, and define S(x), the successor of x, as the number to the right of x in the sequence F. Then the system of 1 and S(x) satisfies the Peano Axioms. And yet the system asserts irregular “truths” like 1+1=1, 3+0=2, and so on.

The Peano axioms make no restrictions whatsoever on the nature of the members of the counting sequence. The axioms hold for a vast variety of sequences, including all the specifiable permutations of the sequence 0, 1, 2, 3…

It’s convenient (for practical reasons) to arrange the numbers in order from smallest to largest, but we don’t *have* to do so.

Yes, it’s true that 2 pebbles + 2 pebbles = 4 pebbles. But that doesn’t refute the system I’ve just presented because mathematics proper is not an empirical science. Mathematical constructs can’t be refuted by experiment.

Also, the idea that there are no necessary, absolute truths (and thus that 2+2=4 is not necessarily true) is not just a position of critical theory or wokeness. It also has strong affinity with the philosophies of empiricists like Hume, Mill and especially Quine. It’s also connected with rationally-defensible recent trends like logical pluralism and logical nihilism.

Another rigorous system with irregular “truths” like Bear+Bear=Rocket:

– Form the sequence F: Car, Hand, Bear, Bike, Rocket, Desk, Ball, Air

– Let Car serve as “0”, Hand serve as “1” and so forth.

Define S(x), the successor of x, as the object in the sequence F. Then the system of 1 and S(x) satisfies the Peano Axioms. And yet the system asserts irregular “truths” like Car+Hand=Hand, Bike+Bear=Desk, and so on.

Doesn’t your Sequence F simply assign new symbols to a universally defined numeral? Then S(x) executes functions on these new symbols using the inherent properties/relationships of the original numerals?

I could teach an alien from space that Bear+Bear=Rocket and they would understand it. But they would understand it in exactly the same way I understand 2+2=4. Therefore, the meaning is not changed and no ‘new’ or ‘irregular’ truth is being told, we are simply changing how we express that universal truth.

PS – I yield to your understanding of theoretical math well beyond my own, but I genuinely don’t see how your example does anything other than change symbology – which seems to usually be how Critical Race Theory goes about it’s business.

The Car, Hand, Bear system doesn’t model the Peano axioms because it’s not clear how to extend it indefinitely. However, it’s a perfectly legitimate formal system, provided it’s well defined and not obviously inconsistent. One could calculate with it, etc. It may not be *useful* but that’s okay because the overwhelming majority of axiom systems (formal systems) in math are, essentially, games for advanced mathematicians that have no practical applications. Useful, empirically true, true within an axiom system, and mathematically legitimate are all different things, and have to be clearly distinguished. In the systems I presented, 2+2=5 is true within the system, and the systems are mathematically legitimate (because they’re just as consistent as the ordinary system). If you learn the rules, you can calculate with them. However, 2+2=5 is not useful, and not empirically true.

As I said above: Yes, the meaning changes. But that’s not necessarily a problem. The meaning of “line” changes between Euclidean and non-Euclidean geometry. In fact, the parallel axioms of Euclidean and non-Euclidean geometry directly contradict each other (similar to how 2+2=4 and 2+2=5 contradict each other), so it’s impossible for both to be “really” true (and for “line” to mean the same thing in both cases). And yet both are mathematically legitimate.

As Hilbert says, the meaning of the fundamental terms in a mathematical system (like “line” in geometry) doesn’t really matter. What matters is the logical relations of the axioms and proofs, and those relations are *syntactical*. Meaning doesn’t matter. For example, consider the logical statement: “X is both true and false.” In orthodox logic, that statement is false. Something can’t be both true and false. Right? But note that the falsity of the statement doesn’t depend on the meaning of X. X could mean be statement at all. The truth status is determined by the syntactic *form* of the statement, not the meaning of X.

A good way to understand it intuitively is that computers can check horrendously complex proofs for accuracy, without having the slightest idea of what the symbols mean. If meaning is so important (especially not illicitly “changing the meaning”), then how is that possible?

Again – thanks for the response. You have good points.

As humans, we define terms in order to communicate efficiently with other people which is an incredibly advance ability. Having common understanding of what words (or terms) mean is really critical for a society to function. There will always be misunderstanding and contradictions in what I say and what someone else takes it to mean. But as long as we’re trying to be on the same, we can still be affective.

In my job, common terms can be misunderstood among engineers. We speak of girders and beams. From my perspective, they are interchangeable, but another engineer in another realm of structural engineering may be very confused if I say beam when they expected me to say girder. The definitions are not formal or strict. A structural element can be both a beam and a girder, or only a beam and not a girder, depending on who you talk to. We can work this out.

The definition of a ‘line’ in mathematics obviously has different forms depending how the mathematician is approaching it. Certainly a mathematician sees a line in very different terms than the average person (I would use the term ‘line’ to describe a squiggle or vector or crisscrosses or arrow which i’m sure wouldn’t fly in advanced math). Similarly, I define ‘concrete’ in very strict ways while most people call all sorts of things ‘concrete’.

But while we won’t always on the same page, there is a massive benefit to trying to be. And likewise, I see tremendous downside to throwing out all meaning and declaring it meaningless, simply because it can be redefined. I also see major problems from intentionally introducing new meanings that aren’t clear, with the intent to obfuscate and create upheaval, which is obviously the intent of the CSJW.

But at the end of the day, we aren’t talking about the meaning of terms. No one will argue about whether a term can have multiple and evolving meanings. We’re talking about the dissolution of the Modern method for establishing truth through vetted, established, scientific methods and being replaced for truth that is simply willed into existence. A la “let’s make 2+2=5 a true statement”. Note, the implication that it may not have been a true statement but let’s make it one simply because we can. This is postmodern critical theory and it’s destructive.

I agree. Rational thought can’t even get started if people have divergent definitions of words. As @CitadelMark said on Twitter: “You can’t have productive discussions with modern liberals because they don’t speak English. They invent their own definitions of common words to accommodate their ideology. Prime example: violence.”

Nevertheless, it seems the public has already consented to the woke assault on the dictionary. “Marriage” was redefined by the woke and people fought it hard. Even Barack Obama was opposed to gay marriage in 2008. But the whole thing blew over, and now we all know better. The public approves of the redefinition.

Similar points can be made about the word “woman.” As Timothy Gowers (Fields Medal winner) pointed out in his Twitter thread, the “2+2=5” debate is a surrogate for the “Caitlyn Jenner is a woman” debate. To paraphrase the Orwell quote: “Freedom is the freedom to say that Caitlyn Jenner is a man. If that is granted, all else follows.” Now, the campaign to redefine “woman” is not as far along as “marriage,” but all signs indicate it will follow the same trajectory. So when we say: “It’s better to not change the definitions of words” it’s easy to interpret that as a dog-whistled attack on the redefinition of “marriage” and “woman.” (Gowers sees it that way.)

So the quandary is this: If you oppose redefinitions, do you oppose the redefinition of “marriage” and “woman”? If so, you’re a bigot, at odds with the enlightened views of the general public. If not, it seems you’re not really opposed to redefinitions. One might protest that “marriage” and “woman” are okay, but the next redefinition is really *going too far*! But that’s what everyone said about “marriage” and “woman,” that it was going too far. But after a few years and a strong propaganda effort, the public changed their mind. The wokes were right all along. So it’s easy to believe that the people objecting to woke definitions are always wrong, and will eventually concede in every case.

Read again the part about axioms.

Not being a mathematician, but observing the gist of these arguments, I would say that those who are comfortable with the destruction of concrete meaning embrace R. D. Laing.

In particular :Rule A: Don’t. Rule A1: Rule A doesn’t exist. Rule A2: Do not discuss the existence or non-existence of Rules A, A1 or A2.” R. D. LAING, Knots

Though they may not like

“Experience is mad when it steps beyond the horizons of our common, that is, our communal sense.” R. D. LAING, “Transcendental Experience in Relation to Religion and Psychosis”, The Psychedelic Review

The other part of this is that you’re talking with people who aren’t speaking the same language as you are – their group has special meanings for “plus” and “equals”, for example. So debating with them is pointless. Communication relies on common and shared meanings for words, when this does not apply, the signifiers may be the same, but the thing signified is radically different. If you adopt their definition – “engage with the theory” – then debate is still pointless, since by adopting their definitions, you’ve also assumed their conclusions, you’ve allowed their framing to dominate yours, and you lose the game. And they’ve been conditioned to see only their framing as valid, by the various techniques cults use. Rarely will you be able to go one on one with them for any length of time, and this is important, because the other person can come up with a thought-stopping cliché which will evoke a conditioned response, and subvert any sort of critical analysis. And critical analysis – putting the two systems of thought side by side and comparing them with respect to the outside world – is derailed.

The thing about this nonsense is that it’s useless, there’s no real-world application for it. 01 + 01 = 10 is a useful statement, 01 AND 01 = 01 is a useful statement, and so on. If 01 + 01 has no determinate value, then it’s useless for much of anything – and that’s the same sort of thing as saying that 2 + 2 doesn’t need to equal 4 – here, 2 + 2 has no determinate value. It’s the end of mathematics as a means of modelling physical reality. There’s no need for academic papers or books on this, although people do produce such things and others get suckered into paying for it. People who teach such tripe should be sacked from university or any other teaching position, because they’re teaching nonsense and corrosive nonsense at that – but that could be said for most of sociology, the so-called “soft sciences”, and “grievance studies”. They ought to be out of a job and on the street. Perhaps the closing of schools and universities can accelerate this process.

And debating nonsense can act to give it an air of legitimacy; the proper response is to call it out for the fraud that it is, stop paying people (usually with tax dollars) to promote it, and kick it to the sidewalk.

Most of these examples are simply redefining terms, e.g. 2+2=11 in ternary … this can be stably converted to 2+2=4 in decimal (or base 5 and up). It’s the stability of the conversion that makes it “objective” in the mathematical sense as I understand.

I don’t even know what to say about the rounding example. 2+2=5 if you’re wrong? Sure, I guess, but that explicitly means that 2+2 does not equal 5, so it’s not a counterexample.

The clock / modular one is interesting, but I don’t see the relationship between that and 2+2=4, which I suppose is the point of this article. Even if the clock starts at 2 and makes 2 rotations to point to 1, that has nothing to do with arithmetic. From position 0, there were 2 partial rotations; then there are 2 more partial rotations for a total of 4 partial rotations. What the hands of the clock point to in this case don’t matter with respect to arithmetic or 2+2 equaling 4.

Of course these obvious points don’t really matter or need defending per se. It’s the at least partially insidious, bad faith, or wishful-thinking critical theory / post modern neo-marxist attempt to create a cultural and perhaps violent revolution that needs to be argued against. Thanks to James for doing that so effectively.

I teach math in a Catholic high school. This whole critical theory is why even Catholic schools are losing their edge. Prior to 1980-ish (I am estimating; it might be 1970), Catholic school education was much guided by The Trivium by Sister Miriam Joseph. That was the education I had. I am pretty sure my nun teachers didn’t have teaching licenses, but they sure had a solid liberal arts education. Things have changed a lot since then.

Fast forward to today.

Catholic-school teachers are required to have a state certification – which means you have to go through Colleges of Education and therefore their Crit/SJW classes.

The only math education classes are all constructivist, which was a mild form of Crit/SJW. It seems they have all gone full SJW, no need for constructivism as a cover anymore.

So the Catholic schools are killing themselves.

Just like people ask about the US, if not us, who? I feel the same way about the Catholic schools. If not us, who?

If 2+2=5 then we have solved the wealth inequality problem since 0 dollars plus 5 dollars equals a million dollars, or any other dollar amount you may be interested in. If you disagree then you are racist. Just argue like that, two can play their game.

Someone should warn them not to try that nonsense in a Vegas casino.

It would bring them a world of hurt.

I’m guessing much of this discussion would seem irrelevant to the 60-70% of Americans who don’t go to college and may want a more hands-on career than a higher math univ. prof. Do you need to know about “other ways of knowing” and different bases to find the right parts to repair a car, measure fabric correctly for a sewing project, do head-counts and use recipes for an event, or make sure a child gets the right dose of medicine? The main thing is to not permit much incursion of CT-based ideas into elementary education, when all students get tools for life (whether or not they’re college-bound).

At some points doing math, I remember thinking/feeling, “Wow, that’s really cool how that all fits together!” It was almost an aesthetic pleasure. Do we want to deprive poor students, who may be the most in need of beauty, joy and the satisfaction of problem-solving in their daily lives, of those very things?

As a college English major and lifetime reader, such things as where/how to put commas and semicolons into sentences have been vitally important to me. Wrongly placed punctuation can convey unintended ideas. There need to be universally agreed-upon symbols and conventions – whether commas and semicolons, or plus signs and the use of base ten – for us to communicate. Too much playing around with them will lead to even more misunderstandings than we have now.

As someone who does creative writing and has worked in public health, I think “different ways of knowing” mostly belong to different spheres – intuition and visceral feelings to poetry, logic and reason to STEM. You also need to use rationality in editing a poem, but we should be wary of letting emotions (or the wrong kinds) invade mathematical reasoning.

I read your entire artical, that is 30 min of time I will never get back. I am no mathematician nor a wordsmith as yourself, but I like that you propagate this artical with over 8000 words to basically tell some people that “try” to think they are smarter or better than others that they are dumb.

I think it’s also to definitively describe what’s going on to prevent the 2+2=5 people from endlessly trying to evade the truth.

What surprises me again and again is that you really do not speak hyperbole and do not straw man these people — but show actual quotes of actually published works (long quotes not taken out of context for that matter) that clearly and explicitly state what you are paraphrasing. It is actually so obvious: You just have to take the critical theorists by their words instead of attempting to interpret a liberal position into their words.

That lets me thinking: Why do so many liberals defend it by steelmaning it, create the motte-argument and forcefully interpret some legitimate liberal criticism of the status quo into it? I mean actual liberals that do not really accept the actual critical theory teaching as shown by the fact that they need to steel man it first, create the motte-argument out of it and forcefully interpret some legitimate (liberal) criticism into it before they accept it.

The same liberals would never do that with a populist or far right “alternative fact”. To the contrary: The same liberals that steel man postliberal, neoidentitarian arguments, “criticism” and narratives and sometimes the underlying doctrines from the critical studies are amongst the first that either accept the strawmaning of a center-right position or create the straw man themselves. In fact most are very eager to pick the meanest interpretation of a conservative argument (or talking point) and buy into outright fakes (like photoshopped screenshots inventing a BS headline on FOX news). They would never steel man a Quanon conspiracy theory in order to interpret a legitimate point into it and instead pick the meanest interpretation of a rightwing – populist concern.

That might not be that much of a problem if the BS gets filtered out in the steelmaning process until only inspiration for legitimate criticism remains. But the legitimate criticism is not the only thing radiating into society. In the end the underlying teaching is at least partially carried with it all the time as in a Trojan horse. But that is not even the worst. What really makes it dangerous is that the steelmaning covers and defends the actual believers, teachers/priests and cult leaders in their march through the institutions. The liberals telling themselves that critical studies believes are just badly phrased deep insights that are more liberal as they sound actively help the actual believers, priest and cult leaders and rise in hierarchies of influence and power.

Oh, totally. The thing I find weirdest is not just straw-manning right-leaning arguments or using the meanest interpretation. The weirdest thing is the joy they get by inventing absurd statements by people they don’t like, and then yucking it up over how stupid their target is for making it (when they actually didn’t).

The earliest example I can recall where I got the sense the matrix was glitching was how they’d laugh at Sarah Palin for saying she could see Russia from her porch. But, Palin (for all her flaws) actually didn’t say that — Tina Fey said it during an impersonation of Palin. But, for some reason a lot of people thought it was funny to joke about how stupid Palin was because she said that, only she literally didn’t say that.

Of course, they do it all the time with Trump. Like telling people to go drink bleach or, more seriously, saying he banned Muslims or colluded with Russia or threatened the President of Ukraine. There’s enough to dislike about Trump without making these things up.

Here’s what the woke want:

https://youtu.be/EHAuGA7gqFU

“Self-interest, cronyism, corruption, and an unstable form of uncivilized might-makes-right that will surely eventually collapse into the more brutal and familiar stable sort in which whomever can kill enough people gets to make the rules”

Yes, and I’ll place my money on the people using weapons built by engineers who know 2+2=4. I mean, no wonder these people feel marginalized – you’re not going to win many military battles rejecting white ideas like 2+2=4.

“Whether or not a truth claim is actually true or false misses the point that a political process leads to making that determination”

This reminds me of something Roger Scrouton wrote about post-modernists: someone who says there’s no such thing as truth is telling you not to believe them; so don’t.

Thanks for the clock metaphora for modulo…

…And all the pain you went through with these well intentioned people

To evaluate the hypothesis 2+2 can equal 5, this simpleton placed 2 fingers next to 2 fingers and counted the total. The total equaled the classic definition of 4. The experiment was repeated with apples. The sum still matched the definition of 4. The experiment was repeated with forks. Same end result. Then, to extend the universality of the underlying arithmetic, 2 forks were placed next to 2 apples. The only way to get the count to sum to 4 was REDEFINING what was being counted to ITEMS (from apples & forks).

At this point, an “ah ha” experience was lived. A way, perhaps the only way, to force 2+2=5 is to REDEFINE 5!!

Redefining words and concepts is foundational to critical theory (CT). When discussing CT with the uninformed or open minded, one needs to make this point; to describe the CT version of a word or concept; and to contrast the CT version with the classic version. Doing so will stop, or slow, the infiltration of CT into society.

When “Literally,” has become a Merriam-Webster defined word as, “Could mean literal, could mean figurative,” and we have someone like the Kardashians to blame for this type of garbage, we are living in a world where no one can be told, “You are wrong.”

I mean how on Earth did, “Irregardless,” get an entry into Merriam-Webster? Who’s horrendous idea was this dumbing down of our society to make the simpletons views be right in order to protect their egos from their ignorance?

I’m such a nerd for calling out every misuse of the term “literally”. It literally blows my mind how often it’s misused. I literally laugh my head off every time I see/hear it. Actually, I literally raise my eyebrow in surprise when I hear it used properly.

Another one I commit mental eye rolls over is the term “I could care less” which is always incorrect. Shrug.

Ah! But at the end of “I could care less” is the phantom “but I don’t,” not added of course. You are supposed to “hear” it like you heard “boy” at the end of “You lie,” during that President Obama State of the Union, whoever that congressman was.

This is typical now, isn’t it? When watching cable news, people accept the attribution of motive for an act or verbal expression even when there is no way to know that. I was always taught – and believed – that you can only assess what is going on by what you see. Apparently, mindreading is now part of the whole SJW/Crit nonsense.

I am more concerned that people are becoming comfortable with this.

amazing.

sending regards from Czech republic.

Nice job. I’ve seen PhD students become the unknowing accomplices for acceptance of these idealogies. When these people start explaining the philosophy behind it you begin to doubt your own beliefs.

The only 3 words one must say when confronted with 2+2=5, White Fragility, Systemic Racism, or Viewpoint Epistemology:

NOT EVEN WRONG!

What do you mean here, please, by “Not even wrong.”

Do you mean you think they’re extremely wrong, or not wrong, or…?

It’s a quote attributed to theoretical physicist Wolfgang Pauli. A friend showed Pauli the paper of a young physicist which he suspected was not of great value but on which he wanted Pauli’s views. Pauli remarked sadly, ‘It is not even wrong’.

The idea is that someone has so misunderstood the question that their answer is irrelevant.

So:

2+2=4 would be right;

2+2=22 would be wrong;

2+2=pineapple would not even be wrong because the individual has clearly not understood the problem and, as a consequence, literally doesn’t know what they’re talking about.

More generally, the statement is used to dismiss theories and claims that are not falsifiable — that is, which can be tested and either supported or shown to be incorrect — and thus not able to be examined or discussed in a rigorous and scientific way. Perhaps the statement may be valid in poetical or rhetorical or psychological way, or it may just be funny, but not in any sort of STEM way.

After all, a joke I like goes:

Q: How many surrealists does it take to change a light-bulb?

A: The fish.

It’s not even wrong, but it is amusing.

It’s true that the symbols in 2+2=4 have a default/normal/common meaning (Peano arithmetics, base-10 symbols) outside of specific and explicit contexts.

Arguing about 2+2=5 is the worst way of expressing that observation, so I infer they are gaslighting/trolling rather than genuinely trying to discuss.

Standards are useful (reducing transactions costs and communication overhead). If someone wants to promote a better standard, they are free to do so. But merely confusing issues wastes everyone’s time (that’s my biggest gripe with wokeness).

You’re right that attempts to show 2+2=5 always involve some change of meaning. However, changing meanings is not necessarily illicit in mathematics. The very argument you marshal against 2+2=5 (“change of meaning”) was deployed against non-Euclidean geometry. The intuition was that: if a line can have two or more parallels through a point not on the line, then somehow the meaning of the word “line” has been deceitfully modified. People like Lobachevsky were accused of playing word games and using the word “line” (supposed to refer to straight lines) to refer to curved lines. It also seemed evident, at the time, that our universe is Euclidean, so appeals to experience (like yours) were made to establish that Euclidean geometry is objectively true, and non-Euclidean geometry is false (although now the consensus is that the universe is non-Euclidean at larger scales).

Hilbert’s view was that, in axiomatic mathematics, it doesn’t matter what the ground-level terms (like “line”) mean. Which makes a lot of sense because logical truth/implication depends on syntax, not on meaning. As Hilbert famously put it, “one must be able to say ‘tables, chairs, beer-mugs’ each time in place of ‘points, lines, planes’.”

https://math.stackexchange.com/questions/56603/provenance-of-hilbert-quote-on-table-chair-beer-mug#56605

It’s straightforward to define a rigorous system S with irregular “truths” like 2+2=5. Let S have the same arithmetic statements as the ordinary system (R) we use every day. Define a function f on the natural numbers so that f(x) = x+1 if x is odd, x-1 if x is even. Now interpret the statements of S so that “a + b = c” means f(a) + f(b) = f(c). For example, the statement “2+2=4” in S means f(2) + f(2) = f(4), i.e., 1 + 1 = 3.

This system S is full of empirically false statements, but it is isomorphic to R (because S and R have the same statements), and S and R are equiconsistent. Mathematicians regard isomorphic systems as *mathematically* identical; i.e., their differences are not of mathematical interest. So S seems to be perfectly mathematically respectable. It just *is* the ordinary system R, except we’re reading ‘tables, chairs, beer-mugs’ for ‘points, lines, planes’ (as Hilbert put it). There doesn’t seem to be any way of rejecting S on *mathematical* grounds, despite its assertion that 1 + 1 = 3. A key point to remember here: empirical truth is not the criteria of truth used in mathematics proper. So you are correct in saying (like J. S. Mill) that 2 + 2 = 4 is true because it’s empirically true (2 pebbles + 2 pebbles = 4 pebbles). Nevertheless, rigorous systems where 2 + 2 = 5 can be defined and defended.

I agree with your broader point, that people are trying to gain power by manipulating word meanings. However, that sort of activism can’t fought by stamping one’s foot and insisting on the “correct” usage of words, as tempting as that is. As Wittgenstein put it: “… a word hasn’t got a meaning given to it, as it were, by a power independent of us, so that there could be a scientific investigation into what the word *really* means. A word has the meaning someone has given it.” Meanings are social, determined by mass popularity, and that’s the ground you have to defeat them on, as difficult as that is.

On the off chance you might be interested, I’ve written a paper that goes into some detail on the principles of definitional conflict.

https://www.academia.edu/39908519/The_Definist_Fallacy_Winning_21st_Century_Political_Battles_by_Orwellian_Redefinition_of_Words

When you say “change in meaning” you are effectively introducing a variable. This is also what’s happening in all of the other examples too and is not shown in their equations, hence, they are wrong.

KK – well developed response.

I’m not a mathematician (a bridge engineer actually) so I spent minimal time studying math theory and obviously much more time studying applied math. But it stands to reason that if you redefine a term and plug that term into the original equation, the only way it makes ‘sense’ is based on the original context.

This may be restating your point but if I assign “Bear” = 2 and “Rocket” = 4 then Bear+Bear=Rocket which is true but meaningless without the context that refers back to what “Bear” and “Rocket” were defined to be in the first place which relies wholly on the fact that 2+2=4.

So in your example of systems S and systems R, systems S only ever makes sense when referencing it’s original definition. While mathematics gives ample space for defining terms in whatever way a creative mind can come up with, the end result will always have to rooted in empirical truth. I’d imagine a theoretical mathematician wouldn’t get far is they simply stated 2+2=5 without providing all the context which in the end shows precisely that 2+2=4 and always will.

It’s a good thing Einstein didn’t just publish his theory with the single statement “F=MA. Signed – AE” because those 4 symbols are completely meaningless (and useless) without context of what each means. And by the way, the truth of his equation depends on the fact that 2+2=4 and always will.

Again – i’m in unfamiliar territory here but I’m guessing your example of S and R actually only work if 2+2=4 is empirically truth. I’m guessing all proofs and theorems (which rarely use numerals) can be traced back to some assumption of empirical truth. But hey – the way we define a proof was set up by white men working to assert dominance over the minority so it’s all a mute point anyway…

Applied math can be empirical of course, but math proper virtually never is. Mathematicians don’t refute each other with experiments. They refute each other with proofs from axioms, and the axioms can be pretty much anything. Even sequences of symbols with no meaning whatsoever. Note that both Euclidean geometry and the two varieties of non-Euclidean geometry are all respectable mathematics, even though their respective axioms contradict each other.

Even an elementary statement like “a geometrical circle exists” is empirically false. The geometrical circle is perfectly circular, to infinite resolution, but no real circle is. The same goes for infinite sets. Mathematicians work with them constantly, but no one has ever experienced one in physical reality. Mill was the great exponent of math as an empirical science, and he’s currently regarded as having been totally discredited by Frege and others. Balaguer’s paper “Mill and the Philosophy of Mathematics: Physicalism and Functionalism” is a good overview if you’re curious.

Thanks for the reply. I realize now that I incorrectly used the term ’empirical’. As empirical properly refers to something observable, what I really meant was something with an inherent logical relationship. The first principles we learn in mathematics is the relationships and functional properties between numerals. We can all relate empirically to the simple equation 2+2=4 but mathematics quickly out-distances our ability to see or relate to it as you note.

**Side note: Or maybe it actually enhances our ability to describe something we otherwise can’t grasp. Your example of a perfect circle is a good one, since we can’t understand what it actually is but we can describe it in mathematical terms. Infinity (or (-) numbers or anything large or small…) is something else we can’t get our minds around but can define and put it to use with math.

But even though math quickly becomes non-empirical, it still relies wholly on established rules, or axioms, such as what it means to sum to numbers. We can, if we desire, disregard these axioms, but if we don’t provide a new axiom (context) we create only gibberish.

My response is over a month late so I don’t know if this will be read, but I would point out that there’s a little carelessness being employed here. It’s subtle and having years of mathematical logic history ingrained in me, it took me a long time to see the problems being embraced in the mathematical community (Retirement seems to allow one to re-examine more closely).

Both John P. and Lindsay have the correct idea of the problem, but they are being hit with imprecision. “Empirical” embodies the idea they seek to use, but of course its use in the philosophy of science (and often appealed to in science itself), carries more than needed. Mill’s idea that math can be empirically justified involves a large conceptual foundation unnecessary to what John P. and Lindsay appeal to.

What Lindsay and John P. are referring to is better captured with the term “perceptual.” And here, they have very good grounds for the basis of their argument. Unfortunately, perception was virtually ignored by Frege, Russell, Hilbert, etc. because of their embrace of the Form/Object conflation (we should call it a fallacy), introduced by Kant.

This fallacy goes back to Plato, but reached it’s pinnacle with Kant in his Analytic/Synthetic distinction. It has variants though (Sense/Reference, Intension/Extension, De Re/De Dicto, etc.).

A very nice, but obscure, paper that outlines this problem was written in the 1960’s by Leonard Peikoff. Unfortunately it was buried in a non-academic publication (some even considered it cultish due to it’s connection to Ayn Rand) so never got much attention. But if one reads it critically on its own, the fallacy comes into the open. “How” we perceive does nothing to the metaphysical status of “what” we perceive, as Kant embraces (and is carried into the Analytic/Synthetic). Please see the paper here:

https://courses.aynrand.org/works/the-analytic-synthetic-dichotomy/

Because “perception” precedes “conception” a hierarchy, and hence a dependency relationship exists in the predicates we use. A nice defense of this is given by David Kelley in his book “Evidence of the Senses” originally published in 1982 (a student of Richard Rorty BTW).

The book illuminates Yale psychologist J.J. Gibson’s work on perception which, unfortunately, was largely ignored because of the dominance of B.F. Skinner, and after his decline, the rise of the computationalist/cognitive psychology of Herb Simon et. al.

It is from perception that axioms are derived. What mathematicians call axioms today aren’t what Aristotle, who coined the term, meant. For Aristotle, axioms have their status as foundational propositions because they are unavoidable in any attempt to deny them–they are tied directly to our perception. As Kelly points out, perceptual judgement is different from perception itself.

This is not how the term “axiom” is used in math today. It is important to understand that I am not using this as an attempt to bring down mathematics, but rather noting a historical change that often obscures, and can be used equivocally, so as to undermine human thought. As a side note, Euclid never called his system “axiomatic” or that it contained “axioms.” He lived in the time of Aristotle, and would certainly have been aware of Aristotle’s “axioms,” and most importantly, what made a proposition an “axiom.” Thus, for Euclid, what was important for his system was that its “definitions,” “postulates,” and “common notions,” were foundational for measuring space.

It is easy to get bogged down in all the well known problems with Euclid’s elements. I am not here to defend it from those criticisms. That detracts from what Euclid’s system represents–a system for which space can be measured without contradiction. This has turned out to be very valuable, and attractive, to mathematicians. Following Euclid, we have systems to create theorems for Rubik’s cubes, state machines, physics, chemistry, etc. The attempts by Hilbert, Russell, Frege, etc. to find one system to “rule them all,” fell spectacularly with Godel’s incompleteness theorem.

This, unfortunately, has led to a rejection of knowledge and truth generally. Truth is only possible with respect to a “Model” (or an axiomatic system), which can only be achieved arbitrarily. Arguments to convince us of this are the incorrect characterization of non-Euclidean Geometry refuting Euclidean Geometry, or at the very least, that our Geometric systems are arbitrary. However, non-Euclidean Geometry is inherently dependent on Euclidean Geometry–meaning it could not be developed without Euclidean Geometry preceding it. The fact that it is internally consistent, or that it translates to Euclidean Geometry does not void this dependency relationship. The same is true of Hilbert’s symbols. The symbols derive their meaning just as John P. states–from the previous system for which they are substituted. Non-standard Analysis shares a similar status in relation to standard Analysis.

The papers you reference depend, when boiled down, on the Analytic/Synthetic fallacy. Thus the definitional fallacy you refer to is itself based on an unaddressed form/object conflation. It should be noted that Aristotle did provide a definitional process which takes account of the hierachal nature of predicates from axioms. If one defines terms in this way, the arbitrary and the equivocal can be avoided.

There is no way to provide the long detailed answer necessary in this post. I only ask that you at least explore the the papers or books listed to get a handle on the criticism missing in most mathematical histories.

Your relation is fine for f(a + b) = f(c) but not for f(a) + f(b) = f(c), in other words your function isn’t well-defined under distribution so by definition you aren’t working within the standard arithemetic so your statement “Let S have the same arithmetic statements as the ordinary system (R) we use every day.” is, trivially, provably false.

Your statements all also completely disregard that all orderings of symbols within any language require closure under representation else there is no such thing as “well-definiteness” and you can’t translate any meaning in any form.

You had to assume that the meaning of the ordering that is represented by the mathematical statement 2 + 2 = 4 was true to construct the words to attempt to disprove your assumption.

This is an obvious complete contradiction.

Wittgenstein be damned, a quote by someone whose name we both know, ie an appeal to authority, does not make a statement or proposition true nor meaningful.

You are conflating representation with meaning.

All formalist approaches to mathematics fail because they disregard that the practice of mathematics is performed.

Abstraction occurs within an interactive reality and conforms to whatever set of principles underly that reality.

Whether or not we can completely list all of those principles within a finite framework at any given point, as the formalist approaches take, is completely irrelevant.

I think it is axiomatic that any culture that can build a bridge knows that 2 +2 = 4.

A corollary is: Any culture that denies 2+2=4 can NOT build safe bridges, tunnels, vehicles or airplanes; functioning energy systems or communication systems … Nor can it yield good healthcare.

Because critical theorists embrace “lived experience”, a solution to the threats they pose to the quality of life is requiring them to:

* Drive vehicles designed only with CT over bridges designed only with CT and through tunnels designed only with CT. Ditto trains.

* Fly in airplanes guided by air traffic control systems designed only with CT

* Rely solely on energy systems designed only with CT

* Rely solely on IT systems ”

* Rely solely on CT trained healthcare providers working out of facilities that rely solely on CT conforming energy, communication & IT systems

If they fail to learn from lived experience, their lives will be shortened by CT.

But on the bright side, those shorter lives will seem much longer and filled with exciting moments of surprise than the drab boring lives lived in worlds where the bridge can be relied on to hold the weight of your car, the elevator hold in place as you enter and exit, the aspirin tablet you swallow contains aspirin not arsenic, and 2 + 2 = 4.

It’s an interesting thought exercise to envision what would actually happen to a functioning society if they ‘woke’ up one day and decided that 2+2 could equal 5 in real-world applications. I reckon what took days for covid to shut down, would be mere minutes in this new, literal application of 2+2=5 world.

I’m thinking the first thing to completely break would be anything tied to finances. That first moment a financial transaction is attempted is when the system would begin folding. Only a matter a moments after that and everything is completely shut down.

Perhaps that’s the woke’s dream scenario. The entire system would have to be rebuilt from the ground up.

I’d say that’s exactly what they want.

“Perhaps that’s the woke’s dream scenario. The entire system would have to be rebuilt from the ground up.”

I think that’s what the people at the core of this genuinely yearn for. Pay attention to all the wild slogans plastered on protest signs like “eat the rich” and “burn the system”. Cain incarnate and that’s something I came to understand by listening to Jordan Peterson explain the horror of people with anti-human motivations like Panzram the serial rapist/killer/arsonist, Pol Pot, Stalin, school shooters, etc.

And who will be there to rebuild if there is no western society left unlike after the fall of the USSR or the surrender of imperial Japan or Nazi Germany? Then we can all be equal in our mass graves and the critical theorists can stand triumphant on the wreckage and horror.

Speaking of “alternative math” and people still daring to assert that 2+2=4, see this splendid short film: https://www.youtube.com/watch?v=Zh3Yz3PiXZw

And now it comes to the corporate world, courtesy of Nike, the tech giants, and a horde of others.

https://www.youtube.com/watch?v=pZy4QXLKHlI

What a world we live in that something like this has to be written!

Thank God (or your parents) you exist, James. I doubt I need to say that no bridge/airplane/architect/computer programmer, etc. is debating this. For surely bringing this up with people (who identify as people) who are trying to build things, serious people would be met with disdain at the very, very least.

Only people who are involved in advocations that don’t actually produce anything, e.g., college professors, philosophers, government employees, etc., can afford to utter this type of absurdity.

Unfortunately, those same people, and the ACLU (Yes, men can get pregnant!), are literally ruining our society.

I don’t know how this is going to end, but I suspect not well. I applaud you for your work and your willingness to put yourself on the firing line. I suspect you find much of this humorous in a maudlin sort of fashion, but I’m also sure it must, as it does with many of us, simply drive you nuts.

The problem is many people, smart people, aren’t aware of what’s happening. My sister, brilliant student/physician who is in her late 60’s thought I was kidding about a lot of this. She still isn’t sure. But, as we know in Washington, math is racist.

Please keep at it for as long as you can, you and people like Peter and Helen are our only hope.

I think you already said how it will end: when the theoretical has a trial by fire in the real world in the bridges, buildings, and computers. Perhaps this will actually when the professor is promised 100,000 annual salary but are informed later that the university can say that the series of symbols “100,000” is actually a quantity represented by nothing.

Either critical theory will be rejected or a dark age will begin (a la Anthem).

Please note that not all college professors are alike. Most of us have a brain and recognize that nonsense for what it is. We college professors that aren’t out to play word games do actually produce something… an educated society.

Shraddha Shirude, in her blog, writes the following about modular arithmetic:

***

We count in modulus all the time! It’s how we measure time! I’ll give you an example: 15(mod 7) = 1mod (7). How can we understand this practically? Well, let’s say the first of the month starts on a Wednesday & there is an event that’s happening on the 15th. What day of the week will the event be held on? Well, each week has 7 days (mod 7), 2 weeks is 14 days, so we are left with a remainder of 1 or 1(mod 7). So we can safely say that the 15th day of the month is a Thursday without actually taking the time to count it out!

***

If she can’t even get this most basic example right (she makes the same mistake twice in the paragraph above), what hope is there that she can impart any useful knowledge to her unfortunate students?

No problem, she’ll just redefine Thursday to really mean Wednesday….

Do none of her students have a calendar on their phone?

You are my freakin’ hero. As someone teaching philosophy at a public community college in California, I am daily utterly astonished at the nonsensical gibberish some of my colleagues are flushing into students’ minds. And as you frequently say, “they really think this way.” And actually, that’s not so much the problem. The real problem is that the majority who don’t think this way don’t understand the implications of thinking and teaching this way and they gleefully jump on opportunities to virtue signal while being totally oblivious to the fact that they are playing with something that’s actually dangerous.

Wow. Thank you. Standing on a foundation of work like this, I think we may yet get a foothold on our climb back to a culture that values reason, equality, good faith debate, and genuine cooperation/ collaboration. I applaud you for your vision, and for the manner in which you’ve managed to convey it. Orwell would be proud. …. also, it’s obvious you’re only doing this to cling to your privileged position in society, to not let anyone else have a spot at the table because you’re revolted by the idea that you might have to share even a meager scoop of gruel with a Trans-POC… OH GOD! They’ve gotten to me too! RUN! RUUUUUN!!!!!!!!!!!!!

It has been said that our increasing knowledge of the science (and the universe) itself, it caused because we “Stand on the shoulders of giants” to see the next discovery and the future improvements.

Critical Theory means to cut off those giants at the knees and leave us staring at the dirt.

Fantastic effort James !! SJWs are humorless bores.

A clear explanation to brief those mathematics accomplices. Excellent. We now need a meme-ified version or an equivalent to a ‘parable of the misled mathematicians’ or somesuch, to be able to swiftly send to newly identified woke-adjacent accomplices – in all the other fields of rationality – as specific fields get attacked – in order to alert the Rational to the dangers of being used in this way.

It would be funny if it weren’t so tragic. It would be tragic if it weren’t so funny.

The best defense against SJWs is to get them to say what they actually mean. On one hand they’re well aware of this weakness and get very slippery when trying to coax it out of them. But of late they seem much more brazen with revealing what’s under the hood. I guess we’re becoming harder and harder to shock.

The issue is that they don’t care about being right in any methodological sense. Trying to argue with them is like trying to play chess with a pigeon. You can’t win, because they don’t accept the rules. The only thing that can be done is simply not giving them any attention (not legitimising them by entering discourse with them).

I do not agree that they should simply be ignored. I think they should get lots of attention all of it in the form of unrelenting ridicule. The value of this would be that eventually people would see that they have no answers and they would then lose any credibility at all for their flawed ideas. They need the Ricky Gervais treatment.

You’re a hero. Thank you for doing this.

The first half was truly painful, James. You’ve outdone yourself, I admire your tenacity.