Math is what mathematicians do.
— fishfry
Oh, and this is a perfect example why it is important: with that you'll give here not only your little finger, but your left leg to the worst kind of post-modernists and sociologists that then can proclaim that math is only a societal phenomenon and a power play that a group of people (read men) do. That all this bull of math being something different, having it's own logic or being something special that tells something about reality is nonsense (or in their discourse, an act in that power play) when it's just what mathematicians declare doing. That's obviously not what you meant, but how can your statements be used is important. — ssu

Wow. I'm kind of stunned. I have no idea what I said that triggered such a strong reaction.

Before dealing with that, let me expand on my remark.

Math is a historically contingent creative activity of humans. What was considered math by the Babylonians was different than what was considered math by the Greeks. When Cantor came up with his revolutionary theory of infinite sets and transfinite numbers, Kronecker famously said, "I don't know what predominates in Cantor's theory – philosophy or theology, but I am sure that there is no mathematics there." Yet today, set theory is a basic part of the undergraduate math major curriculum.

So "math is what mathematicians do," is on the one hand glib, trivial, and superficial; and on the other hand, somewhat deep; since it encapsulates the idea that what is regarded as math changes from one generation to the next; and in the end, math is literally what mathematicians do.

The same goes for art. When abstract art began supplanting representational art, I'm sure critics howled. (I can't claim to know much about art history, but I'm guessing). The French impressionists changed people's ideas about what art is; as did the cubists. So in the end, art is what artists do.

Music is what musicians do. Same combination of triviality and depth. People think they know what music is, then Mozart or Chuck Berry come along to expand and alter people's idea about what is music.

So "mathematics is what mathematicians do" was intended by me to capture the historical truth that what one generation considers radical heresy, the next considers orthodoxy. The universe was Euclidean till it wasn't. Physics used to be about the nature of the universe; now it's "shut up and calculate." Physics is what physicists do.

My remark was anodyne, a truism at once trivial yet expressing the idea of historical progress within a creative discipline.

Ok. And now you react strongly. I wish you'd tell me what you mean.

"... you'll give here not only your little finger, but your left leg to the worst kind of post-modernists and sociologists that then can proclaim that math is only a societal phenomenon and a power play that a group of people (read men) do."

I intended to do nothing of the sort. Can you explain what you mean? Surely you know that what we considered math in 1900 was completely supplanted by 1950; and then again by 2000. "What math is" changes all the time. What's wrong with saying that?

"That all this bull of math being something different, having it's own logic or being something special that tells something about reality is nonsense (or in their discourse, an act in that power play) when it's just what mathematicians declare doing.'

I can read the words, but I honestly haven't got a parser for that language. Can you explain to me what you want me to take from this?


"That's obviously not what you meant,"

Well thanks for at least giving me credit for that.

" but how can your statements be used is important."

Who is doing these terrible things with an anodyne statement like, "Math is what mathematicians do?" And what are they doing?"

I'd think that's the most trivial of truths there is. "Why's that guy building a table out of wood?" "He's a carpenter." "Oh, that makes perfect sense then. He's a carpenter, so he does carpentry."

You take exception? I was literally stunned by your response to what I wrote.

What philosophical taboo have I crossed?

ps -- On the other hand, if you object to my implication that math is a social process, I'll be happy to defend that thesis. Perhaps you've heard of the Mochizuki affair. For the past twelve years the highest-end world class analytic number theorists have been arguing about the validity of a published proof of a famous problem. The current Western consensus is that he doesn't have a proof. In Japan, the consensus is that he does. Math is very much a social process.

For example to the question in the OP it is something. Also do note that it does influence on how people see mathematics and how the field is understood and portrayed. Do people see it from the viewpoint of Platonism (numbers are real), logicism (it's logic) or from formalism (it's a game) or something else? — ssu

Most people are just glad to get out of high school algebra alive. Anyone who cares enough to think about these philosophical issues will certainly not be scarred for life by my little remark.

Who is doing these terrible things with an anodyne statement like, "Math is what mathematicians do?" And what are they doing?" — fishfry

Writing papers in social science. Though I've not seen the specific attitude for maths, I've seen the attitude recently for medicine. Medicine is what medical doctors do, thus making it a principally discursive phenomenon. About words rather than bodies.

If you go looking you can find papers on boolean logic being a colonialist abstraction. I just don't want to go searching for this brainrot again.

It's not clear to me whether you're suggesting that my remarks were not pertinent. — TonesInDeepFreeze

I was only explaining that I preferred not to engage.

But in case you are:

The conversation has had many subjects. You mentioned certain isomorphisms. I was interested in that. My remarks about that don't have to comment on structuralism. — TonesInDeepFreeze

Isomorphisms have everything to do with structuralism. An isomorphism says that two things are the same that are manifestly not the same. That's structuralism.

I'm always amused by this common philosophical example, since Venus isn't a star at all. — fishfry

Just to be clear, the example doesn't assert that Venus is a star.

Isomorphisms have everything to do with structuralism. An isomorphism says that two things are the same that are manifestly not the same. That's structuralism. — fishfry

In set theory, 'isomorphism' is not 'two things are the same that are not the same'. Rather, two things are similar; they have structures that are similar. But it's not the case that different objects that are not the same are the same.

Here's the most trivial example:

<{0} 0> and <{1} 0> are isomorphic but <{0} 0> not= <{1} 0> and {0} not= {1}.

I didn't say isomorphism isn't relevant to discussion about structuralism.

Medicine is what medical doctors do, thus making it a principally discursive phenomenon. About words rather than bodies. — fdrake

Is this not perfectly true? Applying leeches used to be medicine. Now it's not. Removing the perfectly healthy breasts of emotionally troubled 12 year old girls used to not be medicine. Now it is. (Please don't bother to tell me that doesn't happen, I have the facts and figures at hand. Just using an obvious contemporary example).

You know the story of Ignaz Semmelweis. Austrian obstetrician, told doctors to scrub and disinfect their hands before delivering babies to prevent fatal sepsis in mothers. They all laughed at him. "What is the scientific mechanism?" He got increasingly frustrated to the point that his family sent him to an asylum to relax. He got into a fight with the guards and was beaten to death the first week.. Then Pasteur came out with the germ theory of disease and everyone said, Oh yeah old Ignaz was right after all.

So goes scientific progress. A substantially social enterprise. It was Planck who noted that science progresses one funeral at a time. Meaning that the old guard dies off and the young scientists are more open to new ideas.

Is this not perfectly true? — fishfry

It's sort of true. The move also denies that eg 2+2=4 is true. It's just valid as a statement of mathematics. The medical equivalent I saw was that... I think it was heroin wasn't addictive, it was addictive in the context of current medical theory.

@fishfry

The annoying bit isn't recognising that stuff is discursively mediated, the annoying bit is saying that because it's discursively mediated it isn't true, or accurate, or real or whatever. The book about heroin I read made the point about medicine and ontology thusly: "there is no ontological distinction between discourse and reality" - IE, what we say about things and things.

Whereas there is such a distinction for maths objects. You can say that 2+2=5, but it isn't.

I once audited a class with an infuriating social anthropology lecturer who wrote 2+2<4 on the board. But hid he fact he was adding the left two numbers as noise sources in decibels and treating the right as a natural number. That was him, by his reckoning, demonstrating the above point. That 2+2 doesn't have to equal 4.

If you've not interacted with these people I envy you.

It's sort of true. The move also denies that eg 2+2=4 is true. It's just valid as a statement of mathematics. The medical equivalent I saw was that... I think it was heroin wasn't addictive, it was addictive in the context of current medical theory. — fdrake

Science is sometimes but not always a social process.

Then again I could defend the more radical framing. Science is done by people. How could it not be a social process?

Medicine is massively a social process. How did corporations come to control our entire health care system? That's not natural and it's not particularly scientific. The covid response was substantially political, not scientific.

In Orwell's 1984, the fact that 2 + 2 = 4 was shown to be ultimately political. In the end, Winston Smith didn't just agree that 2 + 2 = 5 to stop the pain. He came to truly believe it. State coercion is effective that way.

I don't think you are making your point.

If you've not interacted with these people I envy you. — fdrake

On the contrary. I did interact with these people. They closed down the beach near me during covid. There is no healthier place to be during a respiratory disease epidemic than the beach. That's the day I knew they were insane, and that we were dealing with authoritarian politics and not science.

2 + 2 = 4 isn't always political. It's usually not. But Orwell taught us that it sometimes can be.

I don't think you are making your point. — fishfry

Och, I've made it. I imagine you've never had the pleasure of interacting with these people, so you're able to do the sensible thing and see "science is done by people, how could it not be a social process?" as completely separate from "2+2=4 isn't demonstrably true". Unfortunately it is often held that the fact that some practice is socially or discursively mediated undermines any truth claim in the practice. If that seems totally absurd, yes it is, but it is the attitude your comment resembles and @ssu reacted to.

I'm not imputing that set of beliefs to you. Just highlighting what that phrase could suggest if you read it from a certain angle. But I don't think that is an angle you wanted to suggest, or did suggest.

it is the attitude your comment resembles — fdrake

I resemble that remark?

My sense is that we are only arguing about a matter of degree. You don't entirely deny the social component of science. How could anyone? And I don't deny the objective component of science. iow could anyone?

I don't think the social relativist or postmodern or whatever social critics are getting their cues from me. And there's a lot to be said for their points. We saw rational scientific dissent get crushed by "the science," politics masquerading as science. That made an impression on me. It's more important than ever to distinguish between science and scientism; between rational skeptical inquiry, and authoritarian crushing of dissent. That's one of the major themes of the age in which we live, as in the current argument over free speech and "disinformation."

I'm in favor of rational inquiry and skepticism; and opposed to authoritarian crushing of dissent in the name of scientism and political power. You may have a different sensibility regarding these issues.

And I don't deny the objective component of science. iow could anyone? — fishfry

You cease believing in objective properties, that's one of the steps to the conclusion.

I resemble that remark? — fishfry

You don't. If I read your remark out of context, and didn't know your post history, I could read it that way. But I know you didn't mean it like that.

You don't. If I read your remark out of context, and didn't know your post history, I could read it that way. But I know you didn't mean it like that. — fdrake

I am wondering who these people are that you and @ssu think I'm giving comfort to.

Do you mean cultural relativists, postmodernists, etc.? People who think that objectivity and merit are tools of the cis white patriarchy?

If so, I oppose these people. But they're not waiting for the likes of me to give them encouragement.

I'm big on scientific objectivity. However I'm also aware of the social component of science. And I did recently witness massive authoritarian suppression of legitimate scientific skepticism and dissent in the NAME of science. The political reaction to covid was anything but scientific. Epidemiologists warned that lockdowns were contra-indicated for respiratory infections. And like I say, they closed the beach in my little coastal community. You remember when they arrested some guy paddle boarding by himself on the ocean. That wasn't science, but it was done in the name of science.

So when it comes to the forces of science versus anti-science, I'm firmly on the side of science, objectivity, reason, merit, data, and all that.

But I'm opposed to scientism, and the use of the NAME of science to enforce political, anti-scientific orthodoxy.

Hope that's clear.

But I did want to make sure I understand who you and ssu are referring to. The postmodernists and "merit is racist" types? Those folks, I oppose.

But being someone who often sees too many sides of an issue; I will agree with those postmodernists who say that scientific objectivity and "reason" have often been used by colonizers to oppress the colonized. It's a matter of historical record.

So I do have some intellectual sympathy for the postmodernists in that regard.

Do you mean cultural relativists, postmodernists, etc.? People who think that objectivity and merit are tools of the cis white patriarchy? — fishfry

For me it's a particular set of cultural theory tropes. They're generally working in paradigms like "subtle realism", "new materialism" or the less nebulous actor network theory these days. I can't name any contemporary academic names, a couple of friends' colleagues in academia are full of that stuff, and a few old friends (grad students at the time) and their supervisors bought into that hook line and sinker.

They're generally working in paradigms like "subtle realism", "new materialism" or the less nebulous actor network theory. — fdrake

Those are new to me. Evidently they all have Wiki pages.

I surely didn't intend to give aid and comfort to social theorists whose ideas I've never heard of.

That said, I'll stand behind "Math is what mathematicians do." It's not original with me, I read it somewhere. I never intended for it to be a point of conflict, I though it was harmless. Apparently nothing's harmless these days.

If these ideas are flavors of cultural relativism or postmodernism or whatever, I'll be happy to mildly oppose them. But I'm not dying on any of those hills.

You said you wanted to focus on my ideas. Okay, then. Let me explain my motivation for the thread. Please read the following carefully. If I lose you, say where and why. I promise, it's relevant. — Pneumenon

I did read it carefully. It was interesting. I have some minor remarks but no great insights.

So, I am really really interested in whether or not mathematical objects exist in a mind-independent way. Would there be numbers even if we weren't here? I want to say yes. I think that numbers would be here even if we weren't here. I think that brontosaurus had 4 legs long before we counted them. I can't fathom what it would mean to say that it didn't. So that makes me a Platonist, because I don't think that numbers depend on our minds. At least, not in that way. — Pneumenon

Yes, interesting question. Was 5 prime before there were any intelligent agents in the universe? Hard to say. Isn't the question man-made? But weren't there 5 things? I myself go back and forth on this question. I could argue either way.

But there's a problem with Platonism. If I say that something exists, I need to identify it. If I say, "the Blarb exists", then I need to say what the Blarb is. I need an identity condition that picks out the Blarb and only the Blarb. — Pneumenon

Hmmm, identity conditions again. As you know, in math there are existence proofs that show a thing exists without being able to construct or specify it. Before there were people, the earth existed, but nobody had descriptions or words for it.

This is a counter to Platonism, because it confounds the motivation. Would horses have 4 legs if nobody counted them? I, the Platonist, say yes. But if you ask the me for an identity condition, I'm in trouble. See, if say that the legs of a horse are the set {0, 1, 2, 3}, then I've said that the number of horse legs is the natural number four. But is that even true? What if the number of horse legs is the set of all rationals smaller than 4, i.e. the real number 4? How would we know which one it is? — Pneumenon

Well if you're a structuralist the natural number 4 and rational number 4 are the same thing.

By the way, if there were numbers before there were people, were there sets? Topological spaces? Complex numbers, quaternions? All the high-powered gadgets of modern math?

So it comes out to this:

1. To say that a certain thing exists, you need an identity condition for it.
2. You can't always get those identity conditions for mathematical objects.
3. Therefore, we can't say that mathematical objects exist. — Pneumenon

Well mathematical objects surely don't require "identity conditions," whatever they are. I know you explained them but I'm not sure I believe your definition. But in any event, a Vitali set is a standard example of a set that we can show exists, but we don't know which elements are in it, nor can we specify any particular Vitali set uniquely.

Uh-oh!

My hare-brained solution to that was "maybe there's only one mathematical entity". Math is just a single thing. Because then math can be Platonically real without needing identity conditions. You just say that different mathematical objects are what happens when you analyze that one single object in different ways. — Pneumenon

Well math is one thing, although exactly what it is, is historically contingent. But it has subthings. Algebra and analysis. Real numbers and complex numbers. It's a system as @ssu noted, with many subsystems.

I know that sounds weird. Maybe an analogy will help.

So, for example, you know the duckrabbit? — Pneumenon

This didn't do much for me. It's a simple optical illusion. Or are we back to Clark Kent and Superman again?

It looks like a duck if you look at it one way. It also looks like a rabbit, though not both at once. However, you can't just see it however you want; it's not a duckrabbitgorilla. It's not a duckrabbithouse. It's not a duckrabbithitler. So there are two valid ways of seeing it, but only one can be used at a time. And some ways of seeing it are invalid. — Pneumenon

Ok. Not following your point about math. Math is one thing, though it's hard to say what it is. I'd say math is what mathematicians do, but evidently that anodyne statement got a fair amount of pushback. I believe a basic knowledge of the history of math support the statement.

What if math is kind of like that? — Pneumenon

Like an optical illusion?

There's a single mathematical reality, but it looks different depending on how you analyze it. So if you bring a certain set-theoretic framework, math gives you real numbers, and if you bring a different one, math gives you rational numbers. But the rationals and reals aren't separate, self-identical objects. They're just representations, ways of representing one underlying reality. — Pneumenon

Ok, if this is meaningful for you. Not doing much for me. As Poincaré said, math is the art of giving the same name to different things. That's early pre-structuralism.

Basically, you don't need identity conditions for a representation, because representations don't need to be self-identical in that sense. We can answer the question, "Did horses have 4 legs before anyone counted them?" with "yes" (which is what I wanted). That's because the underlying mathematical reality is Platonic and never changes. But then we have this question: "Was that the rational number 4, or the real number 4, or what?" That's the question that initially flummoxed us. But if there's only one mathematical object, that question is no longer sensible. Represent the number of its legs however you like. You can represent it as rational 4, or real 4, or natural 4. The underlying reality is the same. — Pneumenon

Now you're a structuralist! The natural, rational, and real 4 are the exact same number; even though their set-theoretic representations are quite different. That's because there's a copy of the naturals in the rationals and a copy of the rationals in the reals. So we make those structural identifications, and then we can say that there's only one number 4.

I'll come back to that in a moment. Now for structuralism.

The SEP article on structuralism tells me that there's a methodological kind of structuralism, which is basically just a style of doing math. Then there's a metaphysical structuralism, which is an ontology. The former is a style of mathematical praxis and the latter is a philosophical position.

You seem to waver between methodological and metaphysical structuralism, and it confuses me. — Pneumenon

Ok I should go back to SEP. I've skimmed the article a couple of times but evidently didn't catch this distinction. I was more interested in the distinction of philosophical versus categorical structuralism.

On the one hand, you take a stance like, "I'm no philosopher. I just find this to be an interesting way of doing math". On the other hand, you do seem interested in the philosophical implications of structuralism, e.g. when you said that modern mathematics tells us new things about the notion of identity. And you're on a philosophy forum discussing it, rather than a math forum. — Pneumenon

As I say, I failed to catch this particular distinction in the SEP article, but I'll go back and look.

So let's put the discussion on these questions:

1. Does methodological structuralism imply metaphysical structuralism? Or at least, enable it?
2. Is metaphysical structuralism compatible with Platonism? If so, is it compatible with my idea that there is, in a sense, only one mathematical object? — Pneumenon

Don't know the meaning of the terms and don't have much insight into those questions.

P.S. the SEP article has it,

Along Benacerraf’s lines, mathematical objects are viewed as “positions” in corresponding patterns; and this is meant to allow us to take mathematical statements “at face value”, in the sense of seeing ‘0’, ‘1’, ‘2’, etc. as singular terms referring to such positions. — Pneumenon

I think the structuralist view is that there are mappings, say from 1 to 2 and from 2 to 3, that let us capture the order relations. That might be a little different than Benecerraf's original concept of positions. Category theory is all about the mappings. But categorical structuralism and philosophical structuralism are not the same, and I'm way out of my depth at this point.

By focusing more on metaphysical questions and leaving behind hesitations about structures as objects, Shapiro’s goal is to defend a more thoroughly realist version of mathematical structuralism, thus rejecting nominalist and constructivist views (more on that below). — Pneumenon

Don't understand the referents. No idea what a realist version of structuralism compared to nominalism and constructivism.

[Shapiro] distinguishes two perspectives on [positions in structures]. According to the first, the positions at issue are treated as “offices”, i.e., as slots that can be filled or occupied by various objects (e.g., the position “0” in the natural number structure is occupied by ∅ in the series of finite von Neumann ordinals). — Pneumenon

As I say, the mappings are more important than the slots. But this might represent different views on the matter.

We ask, "Is the number of horse legs the natural number 4 or the rational number 4?". Well, for Shapiro, the number of horse legs occupies a position. That same position is filled by {1, 2, 3} in the naturals and { x ∈ Q : x < 4 } in the reals. So the answer is, "The number of horse legs is both of those". — Pneumenon

Ok. But that is the structuralist view in terms of mappings. The integers have a natural embedding in he reals that preserves all their algebraic and order properties, so we can "identify" them; which is to say, we can consider them the same.

The identity condition, then, is "That unique office occupied by {1, 2 3} in the natural numbers". But there are plenty of other identity conditions that pick out the same object: "That unique office occupied by { x ∈ Q : x < 4 } in the reals" picks it out as well. At this point, the numbers themselves are identity conditions for offices! — Pneumenon

Ok. Nobody says identity conditions are unique. For example in math texts it's common to introduce some mathematical object by listing several different characterizations, showing they're all equivalent, and then giving a name to anything that satisfies any of those conditions.

Maybe I don't need to reduce math to one object after all... — Pneumenon

I think your one-object idea is murky to me. I don't follow it.

I hope I didn't wander off too far. And I hope that this is, at least, interesting. — Pneumenon

Yes, interesting. Can't add much at my end.

Yet today, set theory is a basic part of the undergraduate math major curriculum — fishfry

Perhaps at the more prestigious schools, and maybe at less elite institutions as well. I think I checked on this for Harvard and found such a course, but the branch of the state university where I taught doesn't seem to offer such a course, although set theory is mentioned in a couple of conglomerate classes.

Math is very much a social process. — fishfry

Absolutely. Although many mathematicians work alone much of the time, with social contact allowing critiques by colleagues. It gets very social when one publishes a paper, with regard to referees.

Isomorphisms have everything to do with structuralism. An isomorphism says that two things are the same that are manifestly not the same. That's structuralism — fishfry

I might comment that whereas isomorphisms are very important in mathematics, not all practitioners are heavily involved with them. I have written many papers and notes without mentioning the word. However, the trend for the past how many years, maybe 70 or 100 or so, has been to rise above the nitty gritty of much of classical material and look for generalities that show how one subject in one area is "isomorphic" (use here in a more general sense) to another subject in another area. Or create generalities that when applied specifically to a lower level collection of results show them to be instances of one higher outcome.

I didn't go in this "modern" direction, and my late advisor would speculate that where he and I explored (ground troops, not aviators) would at some future time return to fashion. These days there is such a plethora of subject matter I'm not sure what is fashionable.

And now you react strongly. I wish you'd tell me what you mean. — fishfry

I didn't mean to react strongly. And do note that I said I opposed the kind of thinking. I tried to give the example that in many social sciences (and humanities), the people talking about mathematics (or sciences in general) use simply their own fields discourse and focus, yet then make conclusions of mathematics itself based on these findings. And that's what I referred to giving the left leg to them when saying "mathematics is simply what mathematicians do". That's true of course, math is made by mathematicians, but how that is understood can be quite different.

I should clarify what I mean by this.

The teaching of mathematics in the school system is an educational question, how well the educational system works, not actually about the subject itself, mathematics. Or if as @jgill commented, women make up 25-30% of PhD students in America and 15-20% of math faculties. Thus the make up isn't at all close to the natural 50/50 divide, hence mathematics is male dominated. Yet there being more male mathematicians than female isn't a question about math itself again. It might be a question how faculties work and what kind of groups there are and who has control of the faculty, but that isn't mathematics itself. It would be hilarious to argue the male and female mathematicians would make different kinds of mathematics. There obviously isn't "feminine" or "masculine" mathematics, just as there's no difference in the study of some specific field of mathematics done today in the US, Europe or Asia. "European math" and "Asian math" fit together quite well.

Yet exactly these kinds of differences, who is doing math and where is it done, interest the social sciences. Those questions can be indeed interesting, but they aren't about mathematics itself or the philosophy of mathematics. In areas like literature, art and many fields of human activity etc. there is obviously a difference in where and who does it to the end result. Yet if mathematics is extensively studied in some part of the World or another, I would argue that the pure mathematics would be similar. Yet I know that many would disagree with this, seeing mathematics totally similar to these other endeavors.

In fact @fdrake makes the point perhaps even better than me. Hard to talk about mathematical insights if the other person just focuses that your basically just referring to white European males (that are dead). But the fact is that many people in social sciences are simply so mesmerized by the findings of their own field, that they go too far viewing everything as a social construct, a tool of the society to control people and so on. If mathematics is a social construct, then it can change as the society changes. And here we get back to the discussion of this thread: Is math different? If mathematics is a logical system that studies statements (usually about numbers, geometry and so on), that are true by necessity or by virtue of their logical form, wouldn't this mean then that mathematics is different from being just a social construct of our time?

Hopefully this clarified my position.

I am wondering who these people are that you and ssu think I'm giving comfort to.

Do you mean cultural relativists, postmodernists, etc.? People who think that objectivity and merit are tools of the cis white patriarchy?

If so, I oppose these people. But they're not waiting for the likes of me to give them encouragement. — fishfry

But they do have an effect. Well, It's not like the Catholic Church going against Galileo Galilei and others (or what happened to scientific studies in Islamic societies, that had no renaissance), but distantly it resembles it.

I'm big on scientific objectivity. — fishfry

But I'm opposed to scientism, and the use of the NAME of science to enforce political, anti-scientific orthodoxy. — fishfry

Good luck finding anyone here that doesn't share your views.

But some things are political, just like the response to the COVID pandemic. Lock downs would be the obvious political move: the government has to do something and show it's doing something, that it cares about citizens dying. Taking the stance that Sweden did would take a lot of courage, but there the chief scientific authority was against lock downs, so it was easy for the politicians to do so. How you respond to natural disasters or pandemics is a political decision.

One thing is to keep politics out of things like mathematics. Sounds totally obvious, but we live in strange times...

Thus the make up isn't at all close to the natural 50/50 divide, hence mathematics is male dominated. — ssu

Here's a personal anecdote that may be telling: My PhD class had several women. One dropped out for health reasons, and another was the top student, by far. Shortly after graduation she married a forest ranger and became a housewife.

In the international research clique I joined there were several women, but more men than women. A fairly close colleague, a European woman who had left behind a role as housewife, became the holder of an endowed chair at a major Scandinavian university.

wouldn't this mean then that mathematics is different from being just a social construct of our time? — ssu

Mathematics can be thought of as a structure or system or whatever, but certainly it is not "just a social construct". Sociologists are a little irritating.

But they do have an effect. Well, It's not like the Catholic Church going against Galileo Galilei and others (or what happened to scientific studies in Islamic societies, that had no renaissance), but distantly it resembles it. — ssu

I said math is what mathematicians do. I stand by the remark. I reiterate my literal shock that this anodyne and obvious statement generated pushback from two people. FWIW Galileo got in trouble for insulting his former buddy the Pope. That incident was about politics too.

Good luck finding anyone here that doesn't share your views. — ssu

Maybe.

But some things are political, just like the response to the COVID pandemic. Lock downs would be the obvious political move: the government has to do something and show it's doing something, that it cares about citizens dying. Taking the stance that Sweden did would take a lot of courage, but there the chief scientific authority was against lock downs, so it was easy for the politicians to do so. How you respond to natural disasters or pandemics is a political decision. — ssu

Correct. And calling skeptics "anti-science" is politics too. You're agreeing with me, not disagreeing.

One thing is to keep politics out of things like mathematics. Sounds totally obvious, but we live in strange times... — ssu

Pure math, maybe, though the Mochizuki affair is of interest.

But "mathematics" can be interpreted as the use of data for political purposes. Mathematics is highly political. The NSA employs more number theorists than academia does.

Nothing involving humans is above politics. Professor John Kelley had to leave U.C. Berkeley when he refused to sign a loyalty oath. Fields medal winner Steve Smale was subpoenaed by the House Unamerican Activities committee over his anti-Vietnam war comments. Many other examples could be cited.

Hopefully this clarified my position. — ssu

I appreciate the clarification. It doesn't change my opinion that math is what mathematicians do; nor my surprise at receiving pushback over the matter. Clearly I can't do anything about what various social theorist, postmodernists, et. al. might make of my remark.

If mathematics is a logical system that studies statements (usually about numbers, geometry and so on), that are true by necessity or by virtue of their logical form, wouldn't this mean then that mathematics is different from being just a social construct of our time? — ssu

I used to think so. I probably still do. Still ... Newton and Kant's absolute space and time are reflections of the European paradigm of society in their day. Some philosophers have so argued. I'm not prepared to go into that too deeply. But I don't see anything humans do as above politics, even math. Even if the radical social philosophers are wrong, they're not 100% wrong.

Of course I oppose this kind of thinking when it comes to the math is racist curriculum revisionism these days. Public school math curricula are a disaster. I would say that publicly I agree with the purity of math; but privately, I'm not willing to totally dismiss the critics.

I might comment that whereas isomorphisms are very important in mathematics, not all practitioners are heavily involved with them. — jgill

More of an algebra thing, as is category theory.

(ground troops, not aviators) — jgill

Reminds me of Tim Gowers's distinction between problem solvers and theory builders.

https://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf (pdf link)

Reminds me of Tim Gowers's distinction between problem solvers and theory builders. — fishfry

Somewhat similar, but not quite the same thing. I've never particularly enjoyed solving problems, but rather exploring where certain specific ideas in classical analysis lead. The celestial aviators can cruise the heavens taking us ground troops on ethereal adventures.

Somewhat similar, but not quite the same thing. I've never particularly enjoyed solving problems, but rather exploring where certain specific ideas in classical analysis lead. The celestial aviators can cruise the heavens taking us ground troops on ethereal adventures. — jgill

Sounds interesting. Life in the complex plane. By the way have you seen much of the modern graphing software that's so good at representing complex functions and Riemann surfaces and the like? Don't you wish you'd had that back in the day? I wish they'd had LaTeX, I always had bad writing.

@Pneumenon, I wanted to mention that I made yet another pass over the SEP article and I did sort of see what you're talking about regarding structuralism as a handy way of doing math; versus saying that math really "is" that way, some sort of ontological claim about mathematical objects.

Have I got that right?

Also I realized they talked about the 2-node edgeless graph, which has two absolutely indistinguishable objects. They quoted a paper by a mathematical philosopher named Hannes Leitgeb in Germany, whose name I know because he ran a fabulous MOOC a few years ago on the uses of mathematics in philosophy. Not the philosophy of mathematics; but rather the applications of mathematics in philosophy. We even did the Monty Hall problem, which was when I finally understood it once and for all.

I read his paper, and he talked in depth about the 2-element connectionless graph, and he related it to the famous two-sphere problem. So I had the right instinct. In fact the paper relates Indiscernibles of identicals to mathematical structuralism.

So I just want to say that I have a somewhat better understanding of where you are coming from and what questions you are asking. I'm beginning to understand what this thread is about.

From now on you can assume that I have a more nuanced understanding of your questions, and more of a conceptual framework in which to process them.

I'd have to say that if I have to pick a side, I'm more for structuralism as a handy way of doing math. It's a tool. I don't have any strong ontological feelings. When we study the number 5 or the natural numbers in general, we are treating them as a conceptual primitive. Everyone knows what the natural numbers are.

But I don't care that they're encoded within set theory one way or another; or if they're conceptually put into "slots," and their position in the line of slots determines their nature ... that is, we know the number 5 is the number 5 because it sits in slot 5 in our "slots not set" model of the natural numbers.

I regard that only as a conceptual idea. A proof of concept that if we wanted to, we could encode the natural numbers in ZF, or in "slot theory," if they've worked ou the details. I don't think the number 5 is really a set. I the number 5 is some kind of deep archetype in the human brain ... it's out there somehow. It's not its formal model. It's the abstract thingie "pointed to" by the formal model.

I suppose this makes me a Platonist.

But I think I did make that point earlier. Most mathematicians are Platonists. They conceptualize the mathematical objects they work with as "real," as having an independent existence. A number theorist is interested that 5 is prime and has no interest in how the nature of 5 is modeled by set theorists or philosophers. Likewise the group theorists and complex analysts and everyone else. Most working mathematicians never have a single thought about any of this in their entire professional lives.

Here's a personal anecdote that may be telling: My PhD class had several women. One dropped out for health reasons, and another was the top student, by far. Shortly after graduation she married a forest ranger and became a housewife.

In the international research clique I joined there were several women, but more men than women. A fairly close colleague, a European woman who had left behind a role as housewife, became the holder of an endowed chair at a major Scandinavian university. — jgill

We can all believe this. And since people are mathematicians, they can understand the effect when the dean or the higher ups in the universtity or research establishment simply demand that "there should be more women". The obvious reason is simply viewed as toxic. That women have babies and do still become housewives. My wife wrote and finished her PhD when she was nurturing our first baby. When we had second child, she decided to stay home. My income made it possible.

Of course the present culture is in no way as hostile as was the Catholic Church before (to science in general) or the Islamic religion after the brief spell when the Muslim World upheld Western knowledge. Hence if someone cries after the "decolonization of Mathematics", it actually isn't a threat in any way to the study of mathematics.

I said math is what mathematicians do. I stand by the remark. I reiterate my literal shock that this anodyne and obvious statement generated pushback from two people. — fishfry

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

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

Hopefully you get my point.

But "mathematics" can be interpreted as the use of data for political purposes. Mathematics is highly political. The NSA employs more number theorists than academia does. — fishfry

Good that you used the word "interpreted". It's crucial here.

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.

I used to think so. I probably still do. Still ... Newton and Kant's absolute space and time are reflections of the European paradigm of society in their day. Some philosophers have so argued.. I'm not prepared to go into that too deeply. — fishfry

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.

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.

Sounds interesting. Life in the complex plane. By the way have you seen much of the modern graphing software that's so good at representing complex functions and Riemann surfaces and the like? Don't you wish you'd had that back in the day? I wish they'd had LaTeX, I always had bad writing. — fishfry

No I haven't. And I have little interest in Riemann surfaces. I have used MathType for years with Microsoft Word for writing purposes. For imaging, I have found BASIC is excellent for what I want to do, and have written many math programs. The image of the Quantum Bug on my info page was done with a simple program. Higher, more sophisticated languages seem to be directed toward what is popular in math, and what I do is virtually unknown.

No I haven't. — jgill

Oh. Interesting.

Oh. Interesting — fishfry

Retired for 24 years. Lots of things slip by. Hard enough to persist along the lines of mathematical thought I know about.

Retired for 24 years. Lots of things slip by. Hard enough to persist along the lines of mathematical thought I know about. — jgill

I wish I'd been able to visualize complex functions the way people can these days.

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