But you seem to be using visualization software in your images. They didn't have that stuff when I was in school — fishfry

It's just fairly simple BASIC programming that I enjoy creating. I tried Pascal, Fortran, Mathematica, C++, and one or two others, but by the mid 1990s I returned to BASIC. I use Liberty Basic now. Microsoft's Visual 6 was excellent, but one morning I turned on my computer and it was gone. Instead Microsoft tried to get me to use some new programming language you had to employ at their servers. I've never quite forgiven them. I've written 3D programs, but haven't been happy with them. I'm a 2D guy.

Although I don't subscribe to mathematics being one object, when one looks at specific areas of the subject one can say that one object prevails, and there might be representations of that object that appear disparate. I have dabbled with Möbius transformations in the complex plane for years and still do not understand all the nuances of the subject. They form a group under composition, a procedure described below. Here are three representations: (a) geometric - rotations etc. of the Riemann sphere (b) the analytic form (c) the matrix form.
Question: what is the one object being represented?

Confused really why you would be in "literal shock" and why talk of having pushback. — ssu

I'm a little bit at a loss as to how to respond. I find myself defending a hill that I'm definitely not willing to die on. If it made a difference to anyone, I'd gladly deny, renounce, disavow, and forswear my earlier claim that "Math is what mathematicians do." It was a throwaway line, a triviality, a piece of fluff. I can see that someone could use it to argue for woke math or decolonized math or whatever. So I abjure my former heresy. if it will help.

But you put some thought into your response, so I'll do my best to type in some words. Just please be aware that my heart's not in it. Not a hill I want to die on, not even a hill I want to get a hangnail on.

The statement "Math is what mathematicians do" can be interpreted totally differently by for example social sciences. Totally differently what you mean. I do understand your point, but what I'm trying to say here that all do not share your perspective and they will use a totally different discourse. — ssu

Yes I know these people. How bad has it gotten when Scientific American, of all outlets, publishes Modern Mathematics Confronts Its White, Patriarchal Past.

So I hear you loud and clear on this issue.

I truly don't think any of these people are waiting for the likes of me to give them aid and comfort with an anodyne remark like "Math is what mathematicians do." But if you disagree, I'll withdraw the remark. It's not going to help. Scientific American didn't call to ask my opinion before they published that article.

The conclusion and the counterargument isn't that "If then all mathematicians sleep, is sleeping then mathematics?", no, it's not so easy. — ssu

Well, math is what mathematicians do when they're doing math. An even more mindless slogan, therefore even more easily used by the enemies of rationality and merit, I suppose.

It's that if mathematics is just what mathematicians do, then we just can just focus on the mathematicians as group and in their social behavior and interactions and workings as a group. Because what mathematicians do is what is mathematics, we can take out any consideration of things like mathematics itself or the philosophy of math. What the schools of math disagree on isn't important. I'll repeat it: all you need is to look at mathematicians as a group of people and the behavior and interactions. And in the end some can then talk about "decolonization of mathematics", because the study will notice that it's all about "dead white European males". This is just the way some people think. — ssu

I can't do much about those folks. And I can't censor my opinions (and mindless slogans) just in case one of them overhears me and takes comfort in my words.

It's kind of like in politics. Sometimes there's a candidate with flaws. Some of their supporters deny the flaws entirely. Others admit the flaws but affirm their support for the candidate anyway. I'd be in the latter category. I'll tell the truth even if it undermines my own case. Perhaps I've done that here. So be it.

Hopefully you get my point. — ssu

Not entirely, that last paragraph went over my head a bit. I'll agree that I'd be hard pressed to give a definition of mathematics that transcended historical contingency.

Cryptography and secure communications are important, and it's quite math related. And Wall Street uses quants, quantitative analysts, who do also know their math. Would then mathematics be capitalist? Of course not. I myself disagree with these kinds of interpretations. — ssu

You can't separate math from its uses. If I was making the point that math can be political, I'd agree with myself.

You don't have to, it's all quite simple. Thomas Kuhn came up with the term "scientific paradigm" and note that Kuhn isn't any revolutionary and he doestn't at all question science itself. He's basically a historian of science. It's simply a well thought and researched book that states that basically everybody everybody is a child of their own time, even scientists too. And so is the scientific community, it has these overall beliefs until some important discoveries change the underlying views of the community. And that's basically it. — ssu

Ok. Math is a historically contingent human activity. How is that any better than "math is what mathematicians do?" Maybe anti-racist math is the nex big paradigm.

“Grades and Test Scores Do Not Define
Us as Math Learners”: Cultivating Transformative Spaces for Anti-Racist Math Education [pdf link]

It's always the math "educators" and not the mathematicians promoting this stuff. Of course they started with a "land acknowledgment." I've noticed that they never give their real estate back, though.

So ok, you say I'm giving aid and comfort to these people. But how else should I say that math is a historically contingent human activity? Kuhn's paradigm theory says the same thing. One day someone comes along and changes everyone's view of the subject.

Kuhn is subject to the exact same criticism you level at what I said.

For the philosophy of mathematics or the to the question of just what math is, Kuhnian paradigms don't give any answer and actually aren't important. What is important is the questions in mathematics... that perhaps in the end can get a response like a Kuhnian paradigm shift. So hopefully you still think that way, not only probably. — ssu

I never gave any thought to "what math is." It just like what Justice Potter Stewart said about pornography. "I know it when I see it."

Well that's the best I can do today by way of response. But do tell me if you think Kuhn might be subject to your criticism, for noting that the nature of science changes radically from time to time.

I find myself defending a hill that I'm definitely not willing to die on. If it made a difference to anyone, I'd gladly deny, renounce, disavow, and forswear my earlier claim that "Math is what mathematicians do." It was a throwaway line, a triviality, a piece of fluff — fishfry

Exactly what it was intended to be. How about my previous statement about a mathematician is one who scribbles on paper, curses, then wads the paper up and throws it into the trash. Nobody seemed offended by that. Folksy I guess.

Yes I know these people. How bad has it gotten when Scientific American, of all outlets, publishes Modern Mathematics Confronts Its White, Patriarchal Past. — fishfry

That is a pretty bad article. It paints a picture of an entire profession based on a few incidents.

Thank you for the response. I don't think we have any real disagreement here and I think you got my point.

How bad has it gotten when Scientific American, of all outlets, publishes Modern Mathematics Confronts Its White, Patriarchal Past. — fishfry

Well, that article basically states what this is about: attempt to get job positions. What better way is there to accuse a field of study, mathematics, to itself be "white and patriachial", or whatever. But it works. What can the head of a mathematics department say when accused that there are too few if any women or minorities represented in the staff? Stop hiring your male buddies and follow the implemented DEI rules!

With this short interlude to social discourse, I would like to go back to the actual topic of this thread.

What can the head of a mathematics department say when accused that there are too few if any women or minorities represented in the staff? Stop hiring your male buddies and follow the implemented DEI rules! — ssu

One more slight digression from the original topic. I have been in this position. Rules of Affirmative Action applied and the dean asked for the top three candidates. There was a woman, but no minorities. The dean then placed a minority in with our recommendations. When the time came to decide to make an offer the dean picked the minority. It did not work out well in the long run.

I would like to go back to the actual topic of this thread. — ssu

May I suggest focusing on math objects having several representations (like my example four days ago) and speculating on what the object "really" is or looks like. Or where it lies in a metaphysical sense. To say math is one object is absurd IMO.

I think the one issue in mathematics is that defining a "mathematical object" can be difficult if there are equivalencies, multiple ways of representation of the "object". There's a whole field of mathematics just looking at these similarities, category theory.

I think the one issue in mathematics is that defining a "mathematical object" can be difficult if there are equivalencies, multiple ways of representation of the "object". There's a whole field of mathematics just looking at these similarities, category theory. — ssu

Yes, the similarities don't define the object, however. Is an "object" its' representation picked at random? Or, is there a more metaphysical meaning of the one object having representations? Is there an Object Theory? Just thinking of a way this thread might proceed.

Question from the stands: are individual numbers considered objects? I mean, '7' sure looks like 'an object of thought'. 'How many did you have in mind?' 'Oh, I was thinking 7'. 'Seven? You're sure about that? Mightn't it be six or eight?' 'No, I think it's definitely seven'.

Yes, the similarities don't define the object, however. Is an "object" its' representation picked at random? Or, is there a more metaphysical meaning of the one object having representations? Is there an Object Theory? Just thinking of a way this thread might proceed. — jgill

Let's try an example to clarify this idea:

What would be the mathematical object behind/described by the "well ordering theorem", which can be described as every nonempty set of positive integers contains a smallest member?

How is this object comparable to the axiom of choice? They are basically equivalent to each other. Are they still two different mathematical objects? Do they differ and if they do, how?

Thoughts?

You're conflating non-equivalent theorems.

Theorem 1. The set of natural numbers is well ordered by the standard less than relation on the set of natural numbers.

That does not require the axiom of choice.

Theorem 2. Every set has a well ordering.

That is equivalent with the axiom of choice.

/

I don't know what is being asked by "what is the object behind/described by" a theorem.

Do you want to know what object a theorem is? A theorem of a theory is a sentence provable in the theory. A sentence per a language is a certain kind of sequence of symbols. A sequence is a certain kind of function. And symbols themselves may be taken to be certain mathematical objects.

Though, when I say 'object' I profess no particular metaphysical sense. I only mean 'object' in the ordinary sense of 'a thing' or 'something mentioned' or 'the referent of a pronoun such as 'it'', as I don't profess to be able to explicate that notion more than as a basic presupposition for talking about, well, things, as in "the number 1 is something different from the number 2".

Question from the stands: are individual numbers considered objects? — Wayfarer

Sure.

What would be the mathematical object behind/described by the "well ordering theorem" — ssu

This brings up an interesting question: If two things are equivalent, A<->B, does that mean they represent the same math object? In the example I gave there are two ways of representing the object, Mobius transformation, analytical or matrix, without a non-trivial equivalence argument. The WOT and AofC require a logical argument to verify. Representation theory, in general, includes equivalences.

All of this gets technical and may not be suitable for TPF. Also, set theory quickly moves beyond my levels of proficiency. More appropriate for and

In set theory, equivalence does not imply equality. Here's the most trivial example:

{{0 1}} is a partition of {0 1}. And that partition induces the equivalence relation {<0 0> <1 1> <0 1> <1 0>}. And per that equivalence relation, 0 and 1 are equivalent. But 0 and 1 are not equal.

But, of course, one may posit a different mathematical approach by which certain equivalences imply equality. That's a matter of stipulation.

Though if a metaphysical or philosophical ruling on the question is sought, then that is yet another matter and would not be settled by mere mathematical formulations or stipulations.

You're conflating non-equivalent theorems. — TonesInDeepFreeze

Ok, I perhaps I could have better defined the axiom of choice, but in the latter you get the point.

This brings up an interesting question: If two things are equivalent, A<->B, does that mean they represent the same math object? — jgill

That basically was my question. And I think comes to this thread's main question, because mathematics is quite connected.

Perhaps our questions define what we treat as objects. Could logical mathematics in it's entirety be an object compared to let's say something different like poetry.

If two things are equivalent, A<->B, does that mean they represent the same math object?

Barry Mazur has a really neat paper on this question, and at least parts of it are quite accessible. He ends up advocating (maybe just "showing the benefits of" is a better term) of an approach grounded in category theory.

https://www.google.com/url?sa=t&source=web&rct=j&opi=89978449&url=https://people.math.osu.edu/cogdell.1/6112-Mazur-www.pdf&ved=2ahUKEwjZhPfV0qCJAxW9vokEHYuQCMwQFnoECB4QAQ&sqi=2&usg=AOvVaw0j1f7DfoQP7OKuvRZ37rIU

Mathematics thrives on going to extremes whenever it can. Since the “compromise” we sketched above has “mathematical objects determined by the network of relationships they enjoy with all the other objects of their species,” perhaps we can go to extremes within this compromise, by taking the following further step. Subjugate the role of the mathematical object to the role of its network of relationships—or, a
further extreme—simply replace the mathematical object by this network. This may seem like an impossible balancing act. But one of the elegant–and
surprising—accomplishments of category theory is that it performs this act, and does it with ease.

In a very loose sense, there is a neat parallel here to Roveli's Relational Quantum Mechanics or some forms of process metaphysics. I am less hot on those than I used to be, coming around to views that still include a role for the nature/essence of objects (e.g. Aristotle, or some interpretations of Hegel—things might be defined by their relations, but they are not just collections of atomic relations, rather relations are defined by what a thing is as a whole), or Deely's Scholastic-informed semiotic view of things existing in a "web of relations," which still holds on to "realist" intuitions re essence—a "balancing act." (Well, that's all vague I know, but the paper IS interesting!)

And there is also a neat parallel to St. Maximus the Confessor's philosophy and the Patristic philosophers' conception of number, which I will perhaps return to elucidate if I have more time. But basic idea is that things are not intelligible in themselves (although they do have intelligible natures, logoi). For instance, the idea "tree" is only fully intelligible in terms of other ideas such as the sun and water that are necessary for the tree, the soil it grows in, etc. You cannot explain what it is in isolation. Numbers and figures (following the old division between magnitude[discrete] and multitude[continuous]) are included here, in that they only exist where instantiated, in minds or things, and are not wholly intelligible on their own.

This makes even number dynamic in an important sense. To be fully intelligible, things must exist in the absolute unity of the Logos (Christ as Divine Word, but due to divine simplicity we might say God as a whole as well—on this view the entire cosmos is incarnational).

Anyhow, this sort of relates back to the OP. The idea is that, yes, there is a sense in which everything must be one (i.e. unity in the Doctrine of Transcedentals), but there is obviously also differentiation and intelligibility in the many (the old problem of the "One and the Many").

Another interesting thing is how this relates to knowledge. In Metaphysics, IX 10, Aristotle distinguishes between two kinds of knowledge/truth:

-Asytheta: truth as the conformity of thought and speech to reality (whose opposite is falsity); and

-Adiareta, truth as the grasping of a whole, apprehension (whose opposite is simply ignorance).

Obviously, we follow relations discursively, through asytheta. But then it is by coming to grasp the whole (via adiareta), the principles by which these relations obtain, that we gain a more full understanding (St. Maximus gets at this in Ad Thalassium 60). Likewise, in the Arithmetic Diophantus*, although generally dealing with problems whose principles he cannot discover, makes the case that solutions are virtually present in the principles that will allow for solutions (the "many" contained in the "one," e.g. many shapes, with their own distinct quiddity/whatness, flowing from Euclid's postulates).

*Interesting bit of math trivia, Diophantus, living in the third century, seems aware of Lagrange's four-square theorem, and if he had an actual proof of it then it was effectively lost for 1,500 years (we only have some of his books).

He ends up advocating (maybe just "showing the benefits of" is a better term) of an approach grounded in category theory. — Count Timothy von Icarus

Category theory is beyond my pay grade. It's quite popular (the Wiki page has over 600 views per day - people want to know what it's all about). So far it seems not to have included classical complex analysis. When I look at diagrams what is familiar is composition of functions, of which I am fairly proficient.

Barry Mazur has a really neat paper on this question, and at least parts of it are quite accessible. He ends up advocating (maybe just "showing the benefits of" is a better term) of an approach grounded in category theory. — Count Timothy von Icarus

This is a very interesting paper on the subject. Thank you, @Count Timothy von Icarus! :grin: :up:

Too bad that the basics of category theory aren't taught in school. But then again, the educational system doesn't care much about the philosophy of mathematics or the foundations of mathematics.

Too bad that the basics of category theory aren't taught in school. But then again, the educational system doesn't care much about the philosophy of mathematics or the foundations of mathematics. — ssu

I assume you are speaking of "school" as in "university". It is taught at some elite institutions and some not so high on the scale as well. If you mean elementary or high school the very thought is laughable.

I remember being in a discussion with several mathematicians fifty years ago when category theory came up. The consensus was that it "doesn't do anything". It may not prove any theorems other than those about itself. With set theory one can start from scratch and build a logical system, but CT requires knowledge of the various facets of the category. It's mostly an outgrowth of abstract algebra.

Computer science may be another matter.