So I come with disabilities. — Tom Storm

Likewise, but this:

They may question whether mathematical concepts truly represent universal truths or if they are constructed within specific cultural contexts. — Tom Storm

struck me as inherently plausible as a PM position, but inherently implausible as a serious position per se. Im not sure how it could be argued that natural numbers, for instance, are culture-bound as a concept.

:up: This is the matter I'd like to hear more about from someone with more specialized understanding of the subject. It's in the realm of social constructivism, perhaps. And I guess it leads to a subsidiary question, does supposed universal language of maths have to be such as it is, or could it have taken a different form and had the same results? And in this identified difference, does this point to maths being more arbitrary than we think?

I guess, as a non-PM-ist, I'd just posit that the various 'numeral' systems all represent the same thing and can be read across multi-directionally (between languages) and that gives us reason to think its not the case that its socially constructed, other than the specifical symbolic system in use.

AFAIK, no one, including any p0m0, has ever pointed out a 'culture' wherein mathematics does not work (e.g. "0 > 1" ... "2 + 2 = 87" ... "C = 3 π r" ... "150° triangle" ...) or is inapplicable for time-keeping, drumming-dancing-chanting, farming, buillding megastructures, accounting, navigating, etc. Like bivalent logic (Ibn Sina^^), the universality of arithmetic-geometry (Kant) is inescapable. Whether or not a 'culture' adaptively makes use of elementary / advanced mathematics, however, is another matter all together – perhaps, I suspect, mostly an accident of cognitive anthropological development.


https://thephilosophyforum.com/discussion/comment/692175 ^^

does this point to maths being more arbitrary than we think? — Tom Storm

While I’m no math wiz either, I think (else presume) I know enough about maths to express the following (may I be corrected where appropriate):

Some maths are universal in their semantics (however these semantics might be expressed symbolically, if at all so expressed).

From these universal maths then can and often do get constructed derivations which, as such, often enough don’t consists of the same universality of semantics in that which is derived, but are to some extent constructed.

For instance, the mathematical semantic here expressed by the symbol “1” can only be universal. The symbol “one” here holding the semantic of “a unity” (which can get rather metaphysical when getting into the metaphysics of identity theory). It is a universal not only to all humans but also to all lesser animals that can in any way engage in any form of mathematical cognition.

So something like the semantics to 1 + 1 = 2 can only be universal relative to all sentience that is in any way capable of any mathematical cognition regarding addition.

On the other hand, mathematics which are very advanced derivations of this and similarly universal maths—such as surreal numbers or the mathematics to qubits—will be in part contingent on mathematical factors whose semantics are not universal to all those who can engage in mathematical cognition. Such complex mathematics can then be argued to be in some way constructivist (if in no way speculative) and, thereby, to some extent culture-relative.

For example, the Principia Mathematica (written in 1910) is commonly known to take about a thousand pages to in part formally prove that 1 and 1 is in fact equivalent to 2. No such formal proof occurred previously in human history (obviously, this didn’t prevent humans from successfully applying the mathematics of 1 + 1 = 2). Yet, while everyone has always universally agreed that 1 + 1 = 2, the formal mathematical proof of the book by which this is established is not universally agreed upon without criticism. As one example of this, at least one of the axioms the book uses, its introduced axiom of reducibility, has a significant number of criticism—thereby not being universally apparent in the same way that 1 + 1 = 2 is but, instead, being a best reasoned supposition which was set down as axiomatic.

So, 1 + 1 = 2 is universal and hence not culture relative or in any way socially constructed. The formal proof that 1 + 1 = 2 is however not fully comprised of that which is universal and thereby in no way culture relative or socially constructed—but, instead, can be deemed to be in part constructivist in ways which imply the relativity of some of its mathematical semantics (however these are expressed symbolically).

More directly to the quoted question: The mathematical semantics of 1 + 1 = 2 is in no way arbitrary. But it’s formal mathematical proof in some ways is (albeit yet constrained to reasoned best inferences).

The proper answer to the quoted question should then be relative to those specific mathematical notions implicitly addressed. Overall, the answer is "no and yes," this at the same time but in different respects.

------

P.s. In large part posting this in a want to see if any more formally mathematical intellect would find anything to disagree with in what was here expressed.

They may question whether mathematical concepts truly represent universal truths or if they are constructed within specific cultural contexts.
— Tom Storm

struck me as inherently plausible as a PM position, but inherently implausible as a serious position per se. Im not sure how it could be argued that natural numbers, for instance, are culture-bound as a concept. — AmadeusD

The phenomenologist Edmund Husserl analyzed the historical origin of numeration in terms of the construction of the concept of the unit. Number doesn’t just appear to humans ready-made as a product of nature. It requires a process of abstraction. First one has to recognize a multiplicity, and then ignore everything about the elements that belong to the collectivity except its role as an empty unit. Enumeration, as an empty ' how much', abstracts away all considerations that pertain to the nature of the substrate of the counting. Enumeration represents what Husserl calls a free ideality, the manipulation of symbols without animating them, in an active and actual manner, with the attention and intention of signification.
So rather than a perception of things in the world, counting requires turning away from the meaningful content of things in the world. The world is not made of numbers, the way we construct our perceptual interaction with the world produces the concept of number, and this construction emerged out of cultural needs and purposes , such as the desire to keep track objects of value.

Does that to you then imply that something like 1 + 1 = 2 is constructed within specific culture contexts, such that the quantity "1" is arbitrary rather than ubiquitously universal?

Asking whether math is different in other cultures is like asking whether chess is different in other cultures.

Asking whether math is different in other cultures is like asking whether chess is different in other cultures. — Lionino

Not sure what you mean by this. Chess has a long history and has had changes over time in different cultures. For example:

1200–1700: Origins of the modern game

The game of chess was then played and known in all European countries. A famous 13th-century Spanish manuscript covering chess, backgammon, and dice is known as the Libro de los juegos, which is the earliest European treatise on chess as well as being the oldest document on European tables games. The rules were fundamentally similar to those of the Arabic shatranj. The differences were mostly in the use of a checkered board instead of a plain monochrome board used by Arabs and the habit of allowing some or all pawns to make an initial double step. In some regions, the queen, which had replaced the wazir, or the king could also make an initial two-square leap under some conditions.[64] — https://en.wikipedia.org/wiki/Chess#1200%E2%80%931700:_Origins_of_the_modern_game

↪Joshs Does that to you then imply that something like 1 + 1 = 2 is constructed within specific culture contexts, such that the quantity "1" is arbitrary rather than ubiquitously universal? — javra

I’m not a mathematician either, but I know that there are multiple interpretations of the status and role of the number one (and zero) , including whether it is a basis for all other numbers or whether it is derived. Some argue that the concept of 2 is more fundamental than 1. Theses disputes suggest in a subtle way the cultural basis of concepts of number.

I don't mean chess in its embryonic stages or whatnot. I mean chess today as you can play it in any website.

What defines modern chess is its rules. What defines mathematics is also its rules. You can make up any sort of game, but not all games will be chess. You can make up all sorts of mathematical systems, each with its own rules. A different culture may come up with a different kind of mathematics, but its usefulness and applicability will be different from our mathematics — and if it wants the same applications as our mathematics, it must be our mathematics.

including whether it is a basis for all other numbers or whether it is derived — Joshs

I am not aware of any mathematical system in which 0 is derived from other numbers instead of other way around. ChatGPT told me "Another example is in certain number systems, such as the surreal numbers or the hyperreal numbers. In these systems, 0 may be defined in terms of certain sequences or sets of other numbers, providing an alternative perspective on its construction.", but I don't think that is true.

These are far more abstract conceptualizations than that which I was addressing: the semantic which we, currently, in our culture, symbolize by "1" being universally equivalent to the semantics we convey in English by the phrase of "a unity".

So that "one unity and another unity will be equivalent to two unities" is then a universal staple of all mathematical cognition: in all humans as well as in lesser animals.

Hence, my question was intended to be specific to whether you find the semantic of "a unity"/"1" to be arbitrary and thereby not ubiquitously universal?

AFAIK, no one, including any p0m0, has ever pointed out a 'culture' wherein mathematics does not work — 180 Proof

That's the issue right there isn't it. If there are variations in how maths is done, this does not appear to undermine its capacity to produce consistent results every time.

Enumeration represents what Husserl calls a free ideality, the manipulation of symbols without animating them, in an active and actual manner, with the attention and intention of signification.
So rather than a perception of things in the world, counting requires turning away from the meaningful content of things in the world. The world is not made of numbers, the way we construct our perceptual interaction with the world produces the concept of number, and this construction emerged out of cultural needs and purposes , such as the desire to keep track objects of value. — Joshs

That's what I'm looking for. It's not an easy thing to fully understand.

Some argue that the concept of 2 is more fundamental than 1. Theses disputes suggest in a subtle way the cultural basis of concepts of number. — Joshs

Any thoughts on the unreasonable predictability of maths? Does maths allow us to make any assessment of realism?

P.s. In large part posting this in a want to see if any more formally mathematical intellect would find anything to disagree with in what was here expressed. — javra

Great and thoughtful response: I'll mull over it.

what postmodernism has to say about mathematics. — Tom Storm

I found a link to an old article about a postmodern way of doing math.

"Thus, by calculating that signification according to the algebraic method used here, namely: "

Followed by a conclusion that the erectile organ "..is equivalent to the of the signification produced above, of the jouissance that it restores by the coefficient of its statement to the function of lack of signifier (-1)."

Attributed to the french psychoanalyst Lacan..

There is already a lot of pluralism and "questioning all assumptions," in the foundations of mathematics/philosophy of mathematics, so it's hard to see what a post-modern critique of mathematics would find worth critiquing. I've never seen one, and I've certainly looked in places where they might show up.

That said, there are lots of post-modernist critiques of how mathematics is taught. This makes sense as "mathematical foundations," is simply not something most people care or even know about, and so it's not a good place to "challenge power dynamics," at least not for any sort of social effect. Math classes, however, are an entirely different story.

Post-modern critiques of pedagogy on mathematics run the gambit from the readily apparent ("we should get kids interested in the philosophically and theoretically interesting areas of math and not teach it as 'arbitrary calculations that must be performed to pass tests'"), to the plausible ("math would be more interesting if it applied to real world questions, particularly questions of epistemology and statistics, or probability") to the dubious ("allowing some public school kids to take advanced mathematics perpetuates oppression and hurts society because Asian and European-decended kids currently make up a disproportionate number of students in these classes and colleges and employers like to see math credentials,") to the batshit insane ("we should push the limits of student's creativity by introducing elementary school students to category theory and grounding equality relations in that versus set theory so that they realize the many layered meanings of even the most seemingly self-evident of relations.")

This makes sense as "mathematical foundations," is simply not something most people care or even know about, and so it's not a good place to "challenge power dynamics," at least not for any sort of social effect. Math classes, however, are an entirely different story. — Count Timothy von Icarus

Does your language here suggest that you take post modernism to be a posturing deceit?

There is already a lot of pluralism and "questioning all assumptions," in the foundations of mathematics/philosophy of mathematics, so it's hard to see what a post-modern critique of mathematics would find worth critiquing. — Count Timothy von Icarus

I'm not aware of a maths specific critique. Just taking as the starting point anti foundationalism and the notion that all human knowledge is radically contingent. What does this mean for maths and how do post modernist theorists assess it's reliability and, presumably, its lack of grounding?

This is also how I see it. We can of course debate on what exactly are these rules based on, be it a concept of unity, negation etc. but it looks to me that the absolutely minimal set of concepts is not culturally defined, but something like Kantian, universal categories. When we establish the rules, for example Peano axioms, it is not debatable if those rules won't work (unless of course there's a flaw in the rules). It's another thing if some culture refuses to use a set of rules.

The world is not made of numbers, the way we construct our perceptual interaction with the world produces the concept of number, and this construction emerged out of cultural needs and purposes , such as the desire to keep track objects of value. — Joshs

This seems counter to common sense (other than the first half-line). "enumeration" is an act and you're obviously correct here (just think of roman vs arabic numerals), but "number" is merely the observation of more than one thing at a time. The function of 'maths' is unchanged across any iteration.
The concept of number really isn't different anywhere.

Does your language here suggest that you take post modernism to be a posturing deceit?

By no means. It's just that a lot of people into POMO are very open and vocal about wanting their work to achieve some sort of positive "social change." If this is your goal, the very small and isolated world of mathematical foundations is probably not the place to focus.

Just taking as the starting point anti foundationalism and the notion that all human knowledge is radically contingent. What does this mean for maths and how do post modernist theorists assess it's reliability and, presumably, its lack of grounding?

Challenging mathematics lack of grounding is already a major issue in mathematics. It was the defining historical trend in the field over the 20th century. The deflationary theories of truth that came out of undecidablity, incompleteness, and undefinablity seem in the same wheelhouse (more an inspiration for POMO, or ammunition for it, than possible targets). So, attacking the grounding would be nothing new, whereas attacking the reliability seems extremely difficult if we're not talking about applied mathematics (and if we're talking application then we're generally talking about something else outside mathematics). I mean, is any one going to argue that "given we assume Euclid's axioms, parallel lines never meet," is unreliable? That sort of statement is all about what else is true if the axioms are true (not that the axioms are actually "true"). How could a tautology be unreliable?

Certainly there are lots of critiques about how mathematics is used or appealed to in the sciences, social discourse, and philosophy, but that seems less directly related to mathematics itself.

The way in which mathematics would seem to be most open to attack for being unreliable would be in terms of foundations or application. Application is dealt with vis-á-vis other fields, and foundations is already an open question.

Actually, having written that, I wonder if it might be more accurate to say that mathematics is one of the origin points of the POMO perspective and that it just seems to not be an area of focus today because the relevant critique has already incorporated.

It is not that mathematics differ in every culture -- there is a standardization of mathematics across societies. Just like there is a standardization of engineering across cultures.

But the postmodernists would argue that it is empirically derived. This is how you can argue in favor of a postmodern view. Mathematics has an empirical origin -- not from a universal truth. They are not there to question the veracity of the math methods -- they are there to argue against the objective truth -- (referring to a priori or universal truth).

they are there to argue against the objective truth — L'éléphant

And there's the bumper sticker

What I am interested in is the notion that mathematical knowledge is not inherently objective but is shaped by cultural, historical, and social factors. — Tom Storm

That's the issue right there isn't it. If there are variations in how maths is done, this does not appear to undermine its capacity to produce consistent results every time. — Tom Storm

Cuts right to the core of something that we all assume has to be a core, namely math.

On the one hand:

1 + 1 = 2 is universal and hence not culture relative or in any way socially constructed. — javra

the universality of arithmetic-geometry (Kant) is inescapable — 180 Proof

But on the other hand, maybe:

So rather than a perception of things in the world, counting requires turning away from the meaningful content of things in the world. — Joshs

Some argue that the concept of 2 is more fundamental than 1. — Joshs

First of all, it is too important of a question to answer quickly and easily. And then boom:

Challenging mathematics lack of grounding is already a major issue in mathematics. It's all about what else is true if the axioms are true, how could a tautology be unreliable? — Count Timothy von Icarus

This recognizes the issues at the foundations of math but also fixes "math as math" in itself, as a long-form tautology. From within the tautology of math, there is no room for cultural or historical influence. Or maybe the culture is that of universe, and its history is all time, and the society is the society of minds. Only such influences will produce a math, and because these influences are so simple (universe, mind, all time) that math is so simple and need never change - we've fixed it that way in its own axioms.

And I've just built a POMO language around the same math.

We can drop right back into the question and ask, even with new axioms, would we really have a new math?

I don't think we ever can or will. Math is sort of how we think, not what we think. Math turns whatever we think, objective. It makes objectivity by being math. It is therefore, non-cultural. It is just human.

If you are not understanding '1+1=2' then you are not doing math. If you were to prove '1+1=7' you would be using new words, but needing the same logic and math to demonstrate how this still works. Working itself is the math of it.

It is possible to live a whole human life without any math (the animals do it, probably early man did it). Or you could be raised to think all of math is simply addition and subtraction, and never understand cultures and society's that use multiplication or division. But those worlds where a new conception of math, a postmodern sense, might be said to grow don't address the question head-on. Once there is any math, it will always need a logic, and once there is a logic, it will have a math, and once there is math, it will have words and representations for the same things (representations relative to representations), and once there are words, there will be syntax and logic, and math.

And it's not that we are simply a "rational animal" - minds do other things besides math. But we are an an animal that can do math, and when we do math, we are generating the simple, logical, axiom following, universal. So math ends up objective, as objectivity is its default method.

I suspect that postmodernists talking about mathematics woudl be a dime a dozen. Google supports this.

But a mathematician talking about post modernism... that might be interesting.

Challenging mathematics lack of grounding is already a major issue in mathematics. It was the defining historical trend in the field over the 20th century. — Count Timothy von Icarus

Could be. But no one is claiming PM is entirely original in this.

So, attacking the grounding would be nothing new — Count Timothy von Icarus

I'm sure, but no one is saying it is.

whereas attacking the reliability seems extremely difficult if we're not talking about applied mathematics — Count Timothy von Icarus

If this is what they do. But I don't think it is the reliability as such they would unpack, perhaps more the context of that reliability - the world we assume maths seeks to map and explain. But that is my question - what do they argue in this space?

From Joshs earlier response, it seems that Husserl's phenomenology has a framework for exploring the nature of mathematical objects and structures. It examines ways in which mathematical objects are given to consciousness - an investigation of the ontology of mathematical entities. The old quesion: are mathematical objects mind-independent entities, or are they dependent on human consciousness?

And I suspect some postmodernists coming after this might find that the role of consciousness or, perhaps, the human point of view is what gives maths its power. It isn't that maths is discovered but invented. I'm curious how that this might be laid out. I suspect it will be too technical for a layperson.

Here's one.

But a mathematician talking about post modernism... that might be interesting. — Banno

A conversation between both would be interesting (and perhaps incomprehensible).

This recognizes the issues at the foundations of math but also fixes "math as math" in itself, as a long-form tautology. From within the tautology of math, there is no room for cultural or historical influence. Or maybe the culture is that of universe, and its history is all time, and the society is the society of minds. Only such influences will produce a math, and because these influences are so simple (universe, mind, all time) that math is so simple and need never change - we've fixed it that way in its own axioms. — Fire Ologist

Nice.

I don't think we ever can or will. Math is sort of how we think, not what we think. Math turns whatever we think, objective. It makes objectivity by being math. It is therefore, non-cultural. It is just human. — Fire Ologist

Ok. I'd like to hear what @joshs might say in response to this. It simultaneously suggests that maths is an intersubjective phenomenon but what is the relationship of the reality we map maths too (or visa versa)?

Unless what you’re really interested in is postmodern philosophy itself, you’re probably better off looking at the foundations of mathematics and the regular philosophy of mathematics that isn’t usually labelled postmodern(ist).

When I was learning logic I had a look at Frege, Russell, Hilbert, etc., and found that, as @Count Timothy von Icarus has pointed out, doubts about the basis of mathematics are independent of (and preceded by half a century) what I think you mean by postmodernism in philosophy. One way of putting that is to say that some philosophers of mathematics and foundationally inclined mathematicians were becoming postmodern even before postmodernity. (Alternatively, perhaps these concerns are not postmodern at all but are quintessentially modernist)

So in the philosophy of mathematics you got formalism, intuitionism, and so on, alongside Platonism. Social constructivism too. Here’s an open access paper:

Social constructivism in mathematics? The promise and shortcomings of Julian Cole’s institutional account

This leads me to think that social constructivism/constructionism is not necessarily postmodern in the philosophical sense, even if these distinct approaches are lumped together in the popular imagination.

EDIT: And note that the theory discussed in that paper is based on the social construction theory of John Searle, not usually regarded as a postmodernist.

I'm interested in what I might get from member's responses. As I said at the top, time is limited and I have no education in any of this, so I am just wanting to sift through the various views. My trying to read about maths proper would be like teaching card tricks to a dog.

One way of putting that is to say that some philosophers of mathematics and foundationally inclined mathematicians were becoming postmodern even before postmodernity. (Alternatively, perhaps these concerns are not postmodern at all but are quintessentially modernist) — Jamal

That is definitely an interesting strand which you and the Count have raised.

I'll mull over what's come in so far and see if I need to refine my OP quesion somewhat.

Thanks for the article. Looks interesting. Possibly too technical for me, but I like the thrust of the enquiry.

The idea of 'truth-value realism, which is the view that mathematical statements have objective, non-vacuous truth values independently of the conventions or knowledge of the mathematicians' is I guess what I am am exploring too.

. The deflationary theories of truth that came out of undecidablity, incompleteness, and undefinablity seem in the same wheelhouse (more an inspiration for POMO, or ammunition for it, than possible targets) — Count Timothy von Icarus

Not necessarily. After all, Gödel, the originator of the incompleteness theorems, was guided by his self-declared mathematical Platonism, the belief that humanly-created formal systems are ‘undecidable' only in being incomplete approximations of absolute mathematical truths. Husserl’s phenomenology questions the philosophical naivety on which Godel's theory of the object rests.