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.

Software developers are also making their own lives easier. It's no secret you can ask a chatbot to generate code for a website—and it works pretty well. — Carlo Roosen

Recently debunked. Marginal increase in productivity for junior developers, none for seniors. 41% increase in bugs. "Like cutting butter with a chainsaw." It works but then you have to clean up the mess.

Sorry, GenAI is NOT going to 10x computer programming

You don't say how long you've been following AI, but the breathless hype has been going since the 1960s. Just a few years ago we were told that radiologists would become obsolete as AI would read x-rays. Hasn't happened. Back in the 1980s it was "expert systems." The idea was to teach computers about the world. Failed. The story of AI is one breathless hype cycle after another, followed by failure.

The latest technology is somewhat impressive, but even in the past year the progress has tailed off. The LLMs have already eaten all the publicly available text they're ever going to; now they're consuming their own output. When your business model is violating everyone's copyright claims, you have a problem.

The dot product is a straightforward calculation, where the result increases as the vectors align, meaning the more they point in the same direction, the more likely they are to combine. — Carlo Roosen

Hardly a new idea. Search engines use that technique by dot-producting the word frequency of two articles to see how similar they are.

When the grid goes down..." most investors holding gold investments won't have access to "their" physical gold. Isn't it held in trust in some Fort Knox like depositories? The securities of which are both grid dependent and have some physical blunt force inaccessibilities built into the structures — kazan

Well that's a good point too, especially as a lot of gold investment is in ETFs ... so when the grid goes down, everything goes to zero. You're right.

Gold has some "Internet value" (as problematic as that is) but that (decoration, electronics) has little relation to it's value, the bulk of which derives from it's status as a store of value. Cryptos have acquired this status (during which process it is possible to become rich), like it or not. — hypericin

When the grid goes down, the crypto-heads will discover the difference between gold and crypto as a store of value.

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.

No I haven't. — jgill

Oh. Interesting.

@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.

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.

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)

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.

The world is mystical, nah, miraculous in how it is woven together. — Gregory

On this we fully agree.

Maybe mathematics gives you that sense too. Thanks for the conversation! — Gregory

Likewise, thanks.

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.

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 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.

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.

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.

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.

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.

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.

One might have a notion of the Morning star and a notion of the Evening star and believe that they are two different celestial objects. But believing that they are different doesn't make them different. There is only Venus, which is seen at different times of the day. It's not a different object when seen in the morning than it is when seen at night. — TonesInDeepFreeze

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

Now think of such a new AI running on your phone. — Carlo Roosen

Have you seen Google's new AI-summarized search results? It takes forever. My browser delays for five or ten seconds before putting up a result. It's annoying, and the results aren't any better than what I'd get if I just clicked the top link anyway. Tonight I figured out how to turn it off (someone published a Chrome extension for that purpose) and now my web searches are back to the way they were before, serving up search results instantly. I noticed the improvement right away and I have no desire to switch back. They're like everything else AI does ... even when it's correct and useful, it's like swimming in lukewarm oatmeal. There's just something off about it.

I have a feeling the day after you release your AI, someone will release an extension to turn it off. People find this stuff annoying.

Do you often forget your keys when you leave the house? Doesn't the act of locking the door behind you serve as a reminder? Is that the best use case you can think of to motivate my interest? If you were presenting this to an investor, what would be your elevator pitch? As it stands, you haven't piqued my interest. You say, "... the AI has completely different inputs than us humans," but you gave no examples. I already have access to the Internet, and my phone's sensors don't sense anything my own senses already do. Unless you give it LIDAR like an autonomous vehicle, or an infrared sensor, or a built-in CO2 detector. It's straightforward to hook up all kinds of lab sensors to smartphones. That doesn't strike me as much of a breakthrough.

It could be that you have something more impressive than you described. But remember, you get 30 seconds for an elevator pitch. You didn't motivate me.

tl;dr: I see nothing revolutionary about hooking up lab sensors to an AI. I'm sure people are already doing it. Autonomous vehicles sound like the closest thing to what you're describing. They're equipped with sensors humans don't have.

What definition? I didn't take issue with an definition. — TonesInDeepFreeze

I'm gonna quit while I'm behind here.

I didn't intend my posts to comment on structuralism. — TonesInDeepFreeze

My point. That was the subject of the conversation. So I wasn't sure how to respond to your remarks without going off on extraneous tangents.

Do you mean just a "strict linear ordering with the least upper bound property" or do you mean a "complete ordered field"? — TonesInDeepFreeze

My error if I gave the wrong definition. Which I always have to look up since I always forget it. Sometimes the Archimedean property is mentioned, other times not.

The OP is making some point about mathematical structuralism, which was the context for the discussion of instances. I wasn't entirely sure how your remarks related to that.

Unless I've overlooked something, it seems to me that it's easy to prove that it is not the case that any two strict linear orderings with the least upper bound property are isomorphic: — TonesInDeepFreeze

Isn't this the definition you gave earlier?

Was this post for me?

If I mis-stated the definition, my bad.

Hence the question was stupid, as I assumed. — ssu

I didn't say that at all. Only that calling math a system is far too general. It doesn't tell us anything about math.

It being a logical system would be perhaps more fruitful, but the notion would still be in the set of self-evident "So what?" truths about mathematics. Just what kind of logical systems math has inside it would be the more interesting question. Now when mathematics has in it's system non-computable, non-provable but true parts (as it seems to have), this would be a question of current importance (comes mind the Math truths aren't orderly but chaotic -thread). — ssu

Yes ok.

If so, perhaps the old idea of math being a tautology comes to mind: something being random and non-provable but true is... random and non-provable but true. Yet how do we then stop indoor plumbing and the National Football League being math? That the two aren't tautologies, even if indoor plumbing ought to be designed logically(?) Would it be so simple? — ssu

I suppose you'd need a more specific definition if you wanted to uniquely characterize math. It's tricky, since the nature of math is historically contingent. In the old days they didn't even have negative numbers or fractions, or algebra; let alone the wild abstractions we have today.

I'm not sure how important it is to nail down a particular definition. Math is what mathematicians do.

I read this same argument in Kant recently. He wants mathematics to come from our intuition of the world yet doesn't believe the second antimony must apply to appearance. The only reason you don't want math to fully apply to reality is because you suspect a problem with infinite divisibility, right? — Gregory

Not at all. Infinite divisibility is not in question within math. But there's no evidence for it in physics. That's my only point.

Is not 5 yards minus 3 yards 2 yards? Always, forever? Is not 5 feet minus 3 feet 2 feet? I can get smaller and smaller. There is no reason it should end. — Gregory

In physics there is a minimum distance, below which we can not sensibly apply our laws of physics. There is no infinite divisibility in physics.

https://en.wikipedia.org/wiki/Planck_units

You want math to apply to the world when they build bridges but won't go all the way, saying instead there is some invisible indeterminate line across which we can't do math. — Gregory

No, there's a measurement boundary below which we can't do physics. And it's not something I say. It's something Max Planck said, and that a hundred years of physics has found no exception to.

And you say this without a supporting argument. I don't buy it — Gregory

You deny the Plank length? I don't follow your point at all. Physics is very clear on this matter. Nature has a minimum length, below which we can't reason sensibly about. That doesn't mean that infinite divisibility isn't part of nature; it only means that infinite divisibility is not a part of our best theories of physics.

That's why I say that infinite divisibility is part of math; but as far as we know, and until some future genius not yet born comes up with a new idea, it's not part of physics.

Look, I'm not educated. At all. I have neither a degree nor a high school diploma. I have what Americans call a "GED", and that's it. My job is to sit at home in my underwear and write code. If I tried to show up at an academic conference, they'd kick me out just based on smell. I can make computers go bleep bloop, though! — Pneumenon

ok ...

Let me make sure I understand you. You're saying this:

"There's this thing called identity. Math tells us that identity is contextual. So we should also say that the identity of people is contextual, since math tells us about identity. For example, Clark Kent and Superman are contextual identities." — Pneumenon

You opened by saying you wanted to discuss mathematical structuralism; specifically, that you wanted to oppose it.

I'd be happy if we could focus on your ideas. You clarified what you meant by an identity condition, and I pointed out that it was structural, and seemed to undermine your own thesis. You didn't engage and didn't agree or disagree. I'd find it helpful if we could focus in.

I don't understand. "Lois is unaware of the fact that Clark Kent is Superman. Therefore, Clark Kent is not Superman". That's not what you're saying, is it? If I put on clown makeup and then kill someone, I'm still guilty of a crime, even after I wash off the makeup. — Pneumenon

I made the point that Clark Kent and Superman are two representations, or guises if you will, of the same entity. Just as two isomorphic groups are really the same group. I don't think the Superman analogy bears too much weight. Better to come back to your ideas about mathematical structuralism; since I had in the past some training in the fundamentals of contemporary math, including set theory and a little category theory. Then again I have a pretty good background in Superman comics.

Reading over your other points, I have a hunch about your position. You have used the term "representation" a few times. And you seem to think that, if A and B both represent C, then A and B are identical insofar as they represent C. You seem to identify how something is represented with what it is. — Pneumenon

They're not identical. They're isomorphic. Identity is a tricky thing in this context. Structuralism is all about isomorphism. "Sameness" and "identity" don't really come up.

If you would clarify your thesis -- perhaps recall why you initially started this thread and resurrected it -- we could be more focussed.

In modern math, isomorphism is important. Identity and sameness aren't as important; as as we're seeing, they're kind of slippery.

Two things.

1. Do Clark Kent and Superman represent one thing? If so, what is that one thing?
2. Would horses have four legs if nobody counted them? If so, which four would it be? The natural number 4? The rational number 4? The real number 4? — Pneumenon

I wish we could get back to your ideas about Platonism versus mathematical structuralism. The Superman analogy does not bear too much scrutiny in this context.

In any event, if it wasn't clear, in my previous long post I eventually talked myself out of my own point. The additive group of integers mod 4 and the multiplicative group of the integer powers of the complex number i are indeed distinct groups, and distinct objects in the category of groups; although they are isomorphic. I believe that actually supports a point you were making, so if you like we could go back to that before I confused myself on that point.

fishfry will give a much more sophisticated answer to your question. — jgill

I'll just give the laziest answer possible. I typed "is math a system" into Google. The Google AI responded:

Yes, mathematics can be considered a system, as it is a structured set of rules, axioms, and concepts that are interconnected and used to reason about and describe patterns and relationships, often through symbols and operations; essentially forming a logical framework for understanding abstract concepts — GoogleAI

Perhaps a very stupid question: why isn't Math referred simply to being a system? — ssu

Google AI thinks so. Pretty much everything is a system, from indoor plumbing to the National Football League.

There's a discipline called general systems theory, which is ...

... the transdisciplinary[1] study of systems, i.e. cohesive groups of interrelated, interdependent components that can be natural or artificial. Every system has causal boundaries, is influenced by its context, defined by its structure, function and role, and expressed through its relations with other systems. A system is "more than the sum of its parts" when it expresses synergy or emergent behavior. — Wiki

I just wanted to bother you 'cause I thought you'd be fun to talk to. — Pneumenon

Oh gosh I hope I haven't disappointed you ... now I feel pressure to be entertaining!

I figured you'd just ignore me if you weren't into it. I didn't think you'd start screaming "OH GOD WHY DID YOU CHOOSE ME I WANT NO PART IN THIS MADNESS" like I jumped out of the bushes and throttled you or something. — Pneumenon

Not at all. Just on the one hand, feeling totally inadequate to discuss the philosophical side of this ... Quine and Spinoza for example. And on the other hand, feeling that you are mischaracterizing mathematical structuralism and attacking a strawman.

But if it's good for you it's good for me.

It's okay. You can just ignore me if you want. But I think you'll have fun. — Pneumenon

I'm having fun.

Okay, so some philosophers use the terms "qualitative identity" and "quantitative identity" (or numerical identity). Qualitative identity is where you can't tell them apart. Quantitative is where they're the exact same thing like Superman and Clark Kent. "Quantitative" is kinda misleading, there, so I'll drop these terms. — Pneumenon

I'm more confused than ever now regarding those terms. As modern math shows us, concepts like identical and same and "can't tell them apart" depend on the context. They are not absolute. I'll give examples in what follows.

Instead, I'll use "indiscernible" and "identical". — Pneumenon

Big trouble, as the two sphere example shows.

Two things are indiscernible if you cannot tell them apart. They're identical if they are one and the same thing. So two "identical" twins can be indiscernible. But they're not identical, strictly speaking, cause they're two different people. Do I still sound like raving lunatic? — Pneumenon

No, but the counterexamples come up fast. Especially the twins. Identical twins do not have the same fingerprints. Interestingly when I typed "Do identical twins ..." into my Chrome search bar, it autocompleted "have the same fingerprints," showing that tens or hundreds of thousands of people have asked the same question. So identical twins are not identical and we CAN tell them apart. Forget the twins.

But just take some mathematical examples. Is the natural number 3 identical to the real number 3? They are quite different as sets, but they are exactly the same via the usual embedding maps; which is to say that the reals contain a copy of the naturals that respects all the order and arithmetic information, so we regard the naturals as a subset of the reals, even though set-theoretically, this is false.

Consider the x-axis of the Cartesian plane in high school analytic geometry class (or Algebra II in the US). Consider the y-axis. Are they the same? Clearly not, they're different lines in the plane with only one point of intersection. Yet they are both "copies," whatever that means, of the real numbers. But in set theory there's only one copy of any given set. Where'd we get another copy? Are the x-axis and the y-axis the same?

I don't accept the distinction you're trying to make. Too many corner cases.

An identity condition is a statement that only applies to one thing. The statement "a prime number between 2 and 5" is an identity condition, because it applies only to one mathematical object, which is 3. — Pneumenon

You sure? Does it apply to the real number 3, which has an entirely different set-theoretic representation?

But ok. An identity condition is a statement that uniquely characterizes a mathematical object. Like "the unique Dedekind-complete linearly ordered set" uniquely characterizes the real numbers.

Ok I'll accept that definition. But note that it's a structural condition, not a Platonic one.

I better call this out to make sure you note it.

By your definition, an identity condition is a structural condition, not a Platonic one.

I can construct the real numbers many different ways, but they are all the same real numbers, because they all satisfy the identity condition of being a Dedekind-complete, linearly ordered set.

Am I understanding your meaning of identity condition?

And am I making a valid point that this is a structural condition, which seems to negate your point that identity conditions are the opposite of structuralism?

That sounds very simple, right? But it gets complicated. See below. — Pneumenon

No, not simple. Already complicated, and undermines your point that identity conditions are the opposite of structuralism. If I'm understanding you.

(Apropos of nothing, have you ever read Augustin Rayo's The Construction of Logical Space? You might enjoy it.) — Pneumenon

No, but I'm a huge fan of professor Rayo. For a long time he put on a MOOC called Paradox and Infinity, which I took several times because I love the material so much. He went over omega paradoxes (what we'd call supertasks around here), ordinals, the nonmeasurable set, even gave an accessible proof of the Banach-Tarski theorem. This was in a course aimed at total mathematical novices. Fantastic course. He doesn't do it anymore. Last time the attendance was very low. I think everyone is into machine learning and AI these days, if you can judge by the MOOC offerings.

I think we agree that the set {0, 1, 2, 3} is not identical to {1, i, -1, -i}. — Pneumenon

But we can NOT make that agreement. That statement has no truth value till we declare what category we're in; or less technically, till we specify a context.

If we are group theorists, they ARE identical, because there is only one cyclic group of order 4. This point is even more strict if we are in the category of groups. Then there is literally only one such group, and there is no such thing as a representation or presentation or instance of it.

If we aren't in the category of groups, or if we are just beginning to learn group theory, these are two distinct groups that happen to be isomorphic.

The claim that "isomorphism is identity" is part of homotopy type theory. I'm probably lying through ignorance here but that's my sense of the matter.

So no, we do NOT agree that these two isomorphic groups are identical. In beginning abstract algebra class, they are NOT identical, even as groups. They are "merely" isomorphic.

It's only when we take the more abstract point of view of Category theory that we can't tell them apart.

So no. No agreement on this point until you tell me the context.

And we agree that both of those are instances of the cyclic group with order 4, if they have the right operations defined on them. — Pneumenon

In beginning abstract algebra class, yes. In the category of groups, no, because there is exactly one cyclic group of order 4, and I have no idea what it's "made of." I have no concept of its "elements" or its "operations." It has no internal structure at all. Objects in categories do not have any internal structure. All they have is maps to other objects. This is categorical structuralism.

So, no, we do not agree that those are "instances" of anything except in the most casual sense of perhaps beginning group theory. After all, they are the exact same structure.

Would you say that 4 and 2 + 2 are two "instances" of the same number? Good question.

In fact now that I think of it, I have no idea if I've got any of this right. In the category of groups are these two separate objects that have an isomorphism between them? Or is there only one object that has no instances and no internal structure? I'm not entirely sure.

I should mention in passing that earlier today I ran my eyeballs over the SEP article on mathematical structuralism. They made a clear distinction between the philosophical development of mathematical structuralism via Benacerraf and Putnam, on the one hand; and categorical structuralism via Mac Lane and Grothendieck. So categorical structuralism is a branch or aspect of mathematical structuralism, and shouldn't be identified with it as I've been doing.

Now, here are some questions:

1. Is "{0, 1, 2, 3} under addition mod 4" identical to the cyclic group with order 4?
2. Is "{0, 1, 2, 3} under addition mod 4" an instance of said group? — Pneumenon

In the category of groups I don't think the question makes sense. Any more than if you pressed me on whether 4 and 2 + 2 are each instances of the same thing. Is instance even the right word? Maybe representation, or presentation.

They're only "different" in a figurative sense. I think Clark Kent is identical to Superman at all times. They're both one dude. "Clark Kent" shares a referent with "Superman". — Pneumenon

Easily falsified in-universe. Lois Lane has no idea. It's the glasses, apparently. A little willing suspension of disbelief on that point. How can she possibly not see that they're the same guy? But she can't tell.

I don't think that identity is dependent on context like that. There's nothing you can do to make something stop being identical to itself. — Pneumenon

But of course a thing is identical to itself. That's the law of identity. But recognizing when two things are the same is one of the fundamental problems in mathematics!

Now I'm reminded of Barry Mazur's famous essay (pdf link)

When is one thing equal to some other thing, which delves into some of these categorical issues.

Have I ever done so? — Pneumenon

This was in reference to your straw man argument that structuralists make structure a primitive, or some such. I don't think the question was on point but I could be wrong.

Nah, man. It was just a tangential aside. It's okay. I'm not trying to throttle you. I'm just talking to you. I'm just writing this on my laptop. I'm just under your floorboards. Ominously cracking my long, gnarled fingers. And salivating. — Pneumenon

All good, but I'm at the limit of my knowledge and not sure I'm even telling the truth about a few things.

ps - Poincaré said that "mathematics is the art of giving the same name to different things." Another early instance of pre-structuralism.

In fact see the entire context of Poincaré's quote here ... good reading.

https://ncatlab.org/nlab/show/isomorphism

He even extends the idea to physics.

"The physicists also do it just the same way. They invented the term ‘energy’, a word of very great fertility, because through the elimination of exceptions it established a law; because it gave the same name to things different in substance, but alike in form."

pps -- I'm wrong about there not being separate isomorphic instances of groups in the category of groups. That implies that I'm wrong about a few other things in this post. So in the end I'll agree that the two representations of the cyclic group of order 4 are indeed distinct objects, even in the category of groups.

https://math.stackexchange.com/questions/2041417/are-objects-in-the-category-grp-actually-groups-or-isomorphism-classes-of-groups

ppps -- I have resolved my confusion.

So there's a concept called the skeleton of a category, which contains exactly one copy of each isomorphism class of objects. So in the skeleton of the category of groups, there is only one cyclic group of order 4; and there aren't any variations on its representation.

So if we're in the category of groups, ({0, 1, 2, 3}, +) and ({1, i, -1, -i}, *), both exist as distinct, but isomorphic, objects.

But in the skeleton of the category of groups, they do not exist as separate objects; there's only one such group.

So once again, when asking when two mathematical objects are the "same," the answer is always that it depends on the category. And also -- the structuralist view -- we should not use the word same, but only isomorphic, because isomorphism is the only thing that matters.

Would it be mathematically possible to project an infinite plane unto a "discrete chunk" (to use QM language)? — Gregory

Sure. I'll show the procedure in a moment. But what does that have to do with QM? You continually conflate math with physics and I continually note that this is a category error. Do you mean discrete energy levels? That's not any more mysterious than the discrete natural numbers 1, 2, 3, 4, ...

To me this sounds like a contradiction, — Gregory

I will show the procedure in a moment.

but "discrete space" seems like a contradiction to me as well. — Gregory

Here's the Wiki article on discrete space.

A topological space is discrete if all its points are isolated.

A point is isolated if you can draw a little circle around it that doesn't contain any of the space's other points.

Example of an infinite, discrete space: The integers. Think of the integers on the number line:

......-2...-1...0...1...2...3...4.....................

You can see that around each integer, I could draw a little circle of radius 1/4, say, and that circle would not contain any other integer. So each point of the integers is isolated. And since all of the points of the integers are isolated, we say the integers are a discrete space.

We can make any set into a discrete topological space by simply declaring that every subset of the space is an open set.

Alternately, we can declare the set to have the discrete metric, in which the distance between a point and itself is 0, and between any two distinct points is 1.

With this definition we can make the real numbers into a discrete topological space. I know this is counterintuitive, but it only involves an abstract definition. It's a logic game more than anything else, but it's a fact that we can turn the real numbers into a discrete space simply by giving it the discrete metric.

And then the identity function, which maps each real number to itself, is a function that maps the continuous reals into the discrete reals.

That is, we take two copies of the real numbers. One is given the usual topology, which makes it a continuum; and the other is given the discrete topology, which makes it a discrete space.

Then the identity function from the continuous copy of the reals to the discrete copy maps a continuous set to a discrete set. It's even a bijection, which is extra counterintuitive. But that's one way to do it.

Note that there is no physics involved. This is a purely mathematical exercise.

If it's spatial it has parts. Is discrete defined well in mathematics? Again, they use it in QM. — Gregory

Yes, in math a discrete space is any space where every point is isolated, as I noted.

You might be a little put off by using "space" to mean any old set with some kind of topological structure. It has nothing to do with space as in physics or cosmology. And nothing to do with QM. Just math.

ps -- You asked about mapping an infinite plane onto a discrete space. In that case just do the same trick with two copies of the Cartesian plane, one with the usual Euclidean metric and the other with the discrete metric. It's just a math trick, maybe less to it than meets the eye. But we can definitely map continuous spaces into discrete ones.

I'm just talking to you. I'm just writing this on my laptop. I'm just under your floorboards. Ominously cracking my long, gnarled fingers. And salivating — Pneumenon

Yes ok all good. Having fun. Late now and have some things to do tomorrow morning so I'll get to this in the afternoon.

Thanks for your response. The above video is very interesting but it's minute 2 I'm concerned with. This is how i see all geometric objects, and all objects in general actually. — Gregory

Ok, conformal rescaling. Conformal means "angle preserving." So they're mapping the infinite plane onto a finite disk by projecting it through a sphere.

That's an interesting topic. I'm not sure how it relates to the subject of our conversation but if it's meaningful to you, all to the good.

It's not as if i recoil in horror before matter itself, — Gregory

That's good, since matter is all around us.

but i don't understand why something in mathematics so simple cannot be explained to me as if I were 8. — Gregory

Am I failing to explain a mathematical question?

Perhaps you can phrase your question in a sentence or two, clearly, and I'll do my best to explain.

It might be that I'm not quite understanding your question.

As it is, I don't know what mathematical question you are asking.

Maybe I'm just neurally divergent. I've teased apart the finite from the infinite in an object, and in putting them together I find them contradictory, as have many philosophers in history, Hegel being one of them. Good day — Gregory

Well let me know if I can answer any specific questions.

I want to start with this: I'm defending Platonism, bro. — Pneumenon

Oh I see. In your OP you were stating an objection to it only to knock down the objection. That was not clear to me, and I did not read through this entire four year old thread to discern that. My most humble apologies. Bro'.

The objection based on identity conditions really bothers me, so I want to say find some way to get around it and still be a Platonist. — Pneumenon

You know, you might have missed the mark in tagging me for your resurrection of this thread. I made an offhand remark about Hilbert's famous beer mug quote, and I may have mentioned the turn towards structuralism of latter 20th century math. But I am not proclaiming myself the Lord High Defender of these ideas, nor am I particularly knowledgeable about these matters, nor do I even have any particularly strong opinions about them. You don't like structuralism, that's ok by me. You are a Platonist, so was Gödel, and so are most working mathematicians.

So I am poorly positioned to KNOW anything about these matters, nor particularly CARE about them. That is, I am both ignorant and apathetic about the subject at hand.

That said, I did see a bit of modern math a long time ago; and in particular, I was trained in the doctrine of isomorphism; that is, that two things that have the same structure, may as well be regarded as the same thing, in a given context.

So I believe I can perhaps explain some of these ideas, and put them into mathematical context.

But I can't offer any kind of spirited rebuttal to your Platonism, since "I don't know and I don't care."

With that said, I will plunge in and stumble on.

One more thing ...

The objection based on identity conditions ... — Pneumenon

You've used that phrase a few times, and I should admit that I have no idea what it means.

What is an identity condition? Can you explain what you mean? Can you give me an example or two, say with regard to some familiar mathematical examples like the set of natural numbers, or the cyclic group of order 4, or the Riemann sphere, or any other example you care to name.

I just have no idea what you mean by an "identity condition," and how this idea stands in opposition to the idea of structuralism.

Do you mean perhaps that the two representations of the cyclic group of order 4 I gave earlier should be properly regarded as two separate things? But of course they are. They also happen to have the same group-theoretic structure, so they represent the same group. It's all about the context in which we use the word "same." Nobody is arguing with this. Clark Kent and Superman are the same and they are different. It all depends on the context.

I think it's analogous to Quine's observation about modality: — Pneumenon

Now here you are barking up the wrong tree. I know little of Quine and even less about modal logic. I'm singularly unqualified to even think about what you wrote. I simply do not have any place in my mind to hang these concepts. I plead abject ignorance.

Take, for instance, the possible fat man in the doorway; and again, the possible bald man in the doorway. Are they the same possible man, or two possible men? How de we decide? How many possible men there are in that doorway? Are there more possible thin ones than fat ones? How many of them are alike?
— WVO Quine — Pneumenon

I simply fail to see the relevance to any ideas I may have about mathematical structuralism, modern concepts of isomorphism, or anything else. This is exactly the kind of question that, if it were to appear in a thread on this forum, I'd simply ignore. I have nothing at all to say.

I think this is a deep problem. — Pneumenon

I respect that. I just wonder why you are drawing my personal attention to it. I'm singularly unqualified to have an opinion. I have no knowledge, no interest, and I have no referents in my conceptual scaffolding. I have to just read, ignore, and move on. In some other life I was a big Quine fan and knew something of modal logic. In this life, no.

Deep problems manifest in multiple places – that's how you know they're deep. The fact that something similar happens with modality as with math ought to tip us off that this is not a surface-level concern. — Pneumenon

Perhaps you can give me a mathematical example so I can know what you are talking about.

"something similar happens with modality as with math" -- I just have no idea.

Remember, all I did was toss out the famous beer mug quote. Perhaps you think too much of me.

So how does it manifest for math? Like this: — Pneumenon

Ah! Ok. Thanks. I'll gratefully take the lifeline.

Antiplatonist: "We cannot say that mathematical objects exist in a mind-independent sense. This is because there are no clear identity conditions for them."
Platonist: "Identity is structural. Two mathematical objects are numerically identical iff they are structurally identical."
Antiplatonist: "Here are two objects. They are numerically distinct and structurally identical. Therefore, numerical identity is not structural identity. Therefore, you have not answered my objection." — Pneumenon

I can't relate that to any aspect of math I've ever encountered. You threw me an anchor, not a lifeline.

Give me a mathematical example. Something involving the number 5, say,

Again, you are arguing a thesis -- clearly one you've thought deeply about -- that I just have no knowledge of and little interest in.

Why me? All I did was quote-check the beer mug. I might as well have worn Hilbert's famous hat.

Now if you had specific questions about how the concept of isomorphism is used in math, I could definitely be of help. But I did give you a concrete example, the two representations of the cyclic group of order 4, and you didn't engage.

My solution to this dilemma, per the first post, is simple: no two mathematical objects are numerically distinct. — Pneumenon

What does "numerically distinct" mean? I'm willing to agree that 5 and 6 are numerically distinct.

Past that, I have no idea what you mean. Explain please?

There's only one, Math. Any two apparently distinct ones are just logically valid observations of the same object. A duck and a rabbit, if you like Wittgenstein. Attributes, if you like Spinoza. Emanations, if you like Plotinus.

(I like Spinoza. ) — Pneumenon

Didn't he get excommunicated by the Jewish faith? That's all I know about the guy. I hope you see what a philosophical ignoramus you are talking to. I wish I could engage. And I wish to hell you would give me some concrete mathematical examples of what you are talking about.

You wrote this long post to the wrong person. I can't respond to any of it and I have no idea what you are referring to. But some mathematical examples would help. And when you say you're going to give a mathematical example, you don't. You just give some vague hypothetical dialog that has nothing to do with math as I understand it. I wish I could help. Maybe some of the other participants in this thread can chime in.

I agree, with a caveat.

Black's argument is this: "Two things can be indiscernible, in every single respect, and still numerically distinct". He's saying that two things can be indiscernible in every possible way and still be two things.

My argument is a lot more constrained. I'm saying, "Two things can be indiscernible, in terms of mathematical structure, and still be numerically distinct". And this is proved, IMO, by the graph example. — Pneumenon

Do you mean that the two representations of the cyclic group of order 4 are "numerically distinct" yet have the same group-theoretic structure? If that's what you mean, I heartily agree.

I mean, you're not wrong. — Pneumenon

That's high praise around this place :-)

Everybody agrees that there can be two of something in math. As you note, points differ by their location. Location is structural. Therefore, points are structurally distinct. And I agree with you on that. I'm not saying, "Ha! Here are two different points. Suck it, structuralists!"

What I am saying is, "Here are two objects. They are numerically distinct, but structurally identical. Therefore, structural identity is not numerical identity". — Pneumenon

You are using "numerically distinct" in some kind of crazy way. 5 and 6 are numerically distinct. Past that I have no idea what you mean.

But if you are saying that two mathematical objects may have the same structure yet be presented vastly differently, well duh. That's a commonplace observation that nobody would disagree with.

I think you are misconstruing the hell out of mathematical structuralism; flailing at a straw man; and expecting me to engage or to be an enthusiastic advocate of something or other.

If it is possible to individuate two objects without appeal to structure, then we can find two such objects that are structurally identical. Which sinks structuralism. So the burden of the structuralist is to do math in such a way that you must appeal to structure in order to individuate two objects. — Pneumenon

I couldn't parse that. You are talking in such vague generalities. Give me a specific mathematical example.

But the article about MacLane that I recommended to you actually explained that. It said that we can't call two objects isomorphic until we say which category we're in. So the two representations of the cyclic group of order 4 are the same as groups, and different as mathematical objects.

What is the point of making such a trivial observation that nobody disagrees with?

You did this for points, I think. If points are individuated by location, and location is structural, then numerical identity for points is structural identity. It works.

But this doesn't apply to the graph-theoretic example. You can simply say, "There exists a graph with two vertices and no edges". If any graphs exist, this one does. And now we've individuated two vertices without appeal to structure. The structuralist must show that one vertex has a structural property that the other doesn't. By definition, this cannot be done. — Pneumenon

Ok. Fine. Whatever. Tell me why you think I am the right target for this conversation. I can't hold up my end. I'm defenseless.

You can't make structure itself a primitive notion, by the way. — Pneumenon

Have I ever done so? Are you arguing against Category theory? If structure is not a primitive notion, you must be against abstract algebra. I'm actually laughing as I write this. That is such a naive and inaccurate argument. I just don't know what you mean.

That defeats the point of structuralism. The whole point was to abstract away from particulars and deny intrinsic properties. If you make structure primitive, then you've basically made it an intrinsic property. — Pneumenon

Who made structure primitive? Are they in the room with us right now? You're playing word games. And I gather from your discourse that you don't have much idea what structuralism in math is really about.

One last point: when I say, "Therefore, the two vertices are numerically distinct", I'm speaking ex hypothesi. In fact, I do not think the vertices are numerically distinct. — Pneumenon

What the heck does numerically distinct mean? You mean that one is 5 and the other is 6? I don't know what you mean by that phrase.

For one thing, they're unequal as grade school numbers. For another, they're unequal as sets. And for a third reason, 6 is the Peano successor of 5, and the successor of a number is always distinct from the number.

So you're just mathematically wrong here.

There's only one. Any talk of distinction is just a way of talking about how that one object relates to itself. — Pneumenon

Only one number? What the heck are you talking about? You are making demonstrably false claims. Euclidean 4-space is mathematically distinct from the Lorentzian 4-space of general relativity. They're entirely distinct mathematical objects. They both consist of exactly the same underlying set of points; but one has the Euclidean metric and the other has the Lorentzian metric. That makes them utterly distinct; as distinct as Newton's and Einstein's conceptions of gravity.

You just made the claim that there are no distinct mathematical objects.

Do you stand by that? At least it's a claim you made that I am qualified to dispute.

I'm not sure how to cash that out, exactly. Perhaps I should say: math relates to itself in every logically valid way, for every coherent system of logic. Therefore, every coherent logical system expresses math. — Pneumenon

Which tells us what?

Anyway I find this conversation interesting, but I am not really understanding what you're getting at. I could really use some mathematical examples. Do you really claim that 5 and 6 are not "numerically distinct" mathematical objects?

In any event, the observation that two mathematical objects can be

1) Very different in form and nature; and yet

2) Have the same structure with respect to some abstract properties;

is a commonplace barely worthy of note.

Superman and Clark Kent. The same in one context, utterly different in another.

Engineering claringly uses math as if it applies to reality. — Gregory

Claringly, not a word I know. Typo for something else? Clearly? Clarity?

In any event, many disciplines from physics to engineering to biology to economics to baseball statistics use math as a tool. Math itself is studied by mathematicians for its own sake, without regard for utility. But I think you are conflating using math as a tool to study or model an aspect of reality, with reality itself. An error you will repeat below and that I'll try to place in context.

You seem to be saying there is nothing contradictory about continuums or that there would only be such only if they were in the real world. — Gregory

If standard set theory, ZFC, is consistent, then there is nothing contradictory about the mathematical real numbers; as they are a construction within ZFC.

As Gödel showed, ZFC can not prove its own consistency. So we can never be sure if ZFC is consistent, without assuming the consistency of even stronger axiomatic systems. But if ZFC is consistent, then the real numbers are not contradictory. That's as far as I can go. I can't tell you for sure if ZFC is consistent, and I can't claim with certainty that there isn't some terrible contradiction lurking within our conception of the mathematical real numbers.

I have no idea what's true in an absolute sense about the real world, nor does anyone else. We have some fabulous mathematical models of the real world that predict the outcomes of experiments to ten or twelve decimal places (the magnetic moment of the electron being a famous example). But that just says that we have a pretty good mathematical model. It doesn't tell us what nature is all the way down.

Science is about models, not ultimate truth. I think that's pretty well understood these days, but also widely misunderstood.

So then there is something about physical matter that in its properties is not entirely mathematical as we understand that. — Gregory

Other way 'round, I would say. There is something about mathematics that's not necessarily entirely physical. Mathematics has a true continuum. It's unknown whether any such thing exists in nature.

Mathematics has infinite sets; the natural numbers {0, 1, 2, 3, 4, ...} being an example intuitively plausible to almost everyone. You can "always add one more." But there are no infinite collections in nature, as far as we know.

So mathematics has many objects, ideas, and gadgets, that as far as we know, have no correlate or instantiation in nature.

That may be true, although I would like to hear reasons why some day. Where do we draw the line when applying math to matter? How do we know we've gone too far? — Gregory

We apply math to nature to the extent that it's useful. Riemannian geometry is just the thing for general relativity. Functional analysis of Hilbert spaces is just the thing for quantum physics.

We've "gone too far" when we start believing that our mathematical theories ARE reality, as opposed to merely MODELING reality. As you are consistently doing, and as you are about to do in your next paragraph.

String theory vs loop quantum gravity. One has little points that are really strings (1 dimension in 0 dimension?) And the other discrete space. The biggest question in physics (quantum gravity) wants to settle the question of the continuum. They don't want to just throw their hands up — Gregory

This is exactly wrong. In fact you mentioned this wrong idea a while back, in a post that was filled with so many misconceptions and errors (IMO of course, nothing personal) that I didn't bother to answer it, lest I appear to be piling on. Perhaps I should get to it.

String theory and LQG are competing mathematical models, I repeat, mathematical models, that are proposed to be able to predict the results of experiments that we can carry out with our historically contingent experimental apparati.

I don't think anyone claims that they are competing metaphysical theories, claims that nature is "really" that way. Or if they do, they are making a category error, confusing physics for metaphysics.

There is not a dispute between string theory and LQG as to how nature "is." There are two competing ideas for how nature should best be modeled.

Or to put this another way: To the extent that physicists argue about the best mathematical model of the world, they are doing physics. To the extent that they argue about how the world really "is," they are doing metaphysics.

Physics is not metaphysics. Physics is about models. Metaphysics is about the (possibly unknowable by we finite, fallible humans) way things really are. Plato's cave and all that. All we can see is shadows.

I hope this point is clear. Physics is about building models that explain the experiments we can do, up to the limits of precision that we can measure. It is NOT any longer the theory of what is "true" in any absolute sense. This is summed up in the famous phrase, shut up and calculate. Meaning, don't ask what's truly going on. Just use the theory to analyze and predict the results of experiments.

We should remember not to confuse the map with the territory.

Four years later, I had a whim to come back here. I just wanted to explain why this is wrong. — Pneumenon

Ah the good old daze ...

Take graph theory. I show you a graph with two vertices and no edges. By hypothesis, the two vertices are two different things. Those two vertices, however, are structurally indiscernible. Which makes them the same vertex, according to structuralism. Contradiction. — Pneumenon

This is the famous two spheres argument against the identity of indiscernibles.

Max Black has argued against the identity of indiscernibles by counterexample. Notice that to show that the identity of indiscernibles is false, it is sufficient that one provide a model in which there are two distinct (numerically nonidentical) things that have all the same properties. He claimed that in a symmetric universe wherein only two symmetrical spheres exist, the two spheres are two distinct objects even though they have all their properties in common. — Wiki

If this resonates with you, all to the good. I think this is the same argument you are making.

Therefore, either mathematical objects are not identified by their structure, or the stated graph can't be defined. The latter is false. So mathematical objects are not identified by their structure. — Pneumenon

I don't think anyone is saying there can't be two of something in math. There's only one set of real numbers, defined axiomatically as the unique (up to isomorphism) Dedekind-complete ordered field.

Yet we have no trouble taking two "copies" of the real numbers, placing them at right angles to each other, and calling it the Cartesian plane. Or n copies to make Cartesian n-space.

For that matter we say that a line is determined by two points. But one point is exactly like any other, except for their location, yet nobody thinks there's only one point. There are lots of points. And if we wanted to have this conversation, we could argue that there aren't any points at all, a point being a "zero-dimensional location in space," whatever that ultimately means.

Still, two points that aren't the same point determine a unique straight line, and nobody from Euclid to the present has every argued otherwise.

I don't know if this is a violation of structuralism. Structuralism is about deeper matters, not about whether there can be two of the same thing. And if they're the same, how can there be two of them? I feel @Metaphysician Undercover about to take me to task. I don't know the answer to any of this.

That's the fundamental problem with structuralism. You cannot escape the need for identity conditions by focusing solely on relations rather than particulars, because relations are particulars. "I don't have to count objects, because I go by kinds" – and if I ask how many kinds there are...? — Pneumenon

I'm not the self-appointed defender of structuralism. As I say, structuralism is about deeper matters. There is only one set of real numbers, even though there are many ways to define the real numbers.

The set {0, 1, 2, 3} with addition mod 4, and the complex numbers {1, i, -1, -i} with complex number multiplication, are isomorphic as groups. We say they are two different instances of the "same" group even though they are manifestly not the same thing. There is only one cyclic group of order 4, even though we have here two vastly different instances of it. Like Superman and Clark Kent. The same, but very much not the same.

I note that you opened this thread with

A challenge to Platonism, which is IMO one of the more serious ones, is that mathematical objects lack clear identity conditions. — Pneumenon

So are you arguing against Platonism? Or constructivism? They're different philosophies, right? And perhaps even my Hilbert beer mug quote was a little off the mark, since AFAIK he was explaining how axiomatics should work, and not specifically structuralism.

Or perhaps Hilbert was a pre-structuralist. Saunders MacLane, the originator of Category theory, attended Hilbert's lectures. I found a fascinating-looking article, "Saunders Mac Lane: From Principia Mathematica through Göttingen to the Working Theory of Structures" in an anthology called, "The Prehistory of Mathematical Structuralism." So perhaps I intuited more than I realized when I mentioned the beer mug remark. Hilbert was thinking about structures all along.

Perhaps you can glean some clues from this essay, which I haven't yet read but which mentions Carnap, Husserl, and other philosophical luminaries of that era. Evidently these ideas were in the air in the 1930s.

https://academic.oup.com/book/41041/chapter/349348878

(edit) -- I skimmed through the article. Not too many clues in there of an elementary nature directly on point to your concern. In any event, MacLane learned from Hilbert; and Hilbert was indeed a pre-structuralist. The beer mug remark goes deep.

Some things obviously apply to the world. It is often said that there are no perfect shapes in the world. But we can mentally draw a perfect shape WITHIN any object although there it is surrounded by OTHER matter. The shape does exist as a part of another thing — Gregory

If I adjusted my statement to say that the mathematical real numbers nave no necessary bearing on the world, would that be better?

We don't know whether there's a real number-like continuum in nature. Is that better?

Symptoms In Need Of A Cure — Art48

Nice troll bait. I'll play.

There are plenty of anti-science types on the left. Look at the woo community, the astrologists, spiritualists, believers in crystals, etc. Mostly leftists. Marianne Williamson is a Democrat.

And you know you can't get elected to high office in the US without professing belief in God and at least showing up in church from time to time. Obama survived his Reverend Wright scandal ("God damn America!") but he'd never have gotten elected if he was a proud atheist. He may have gone to an America-hating church, but at least he went to church.

Then there's the Reverend Martin Luther King, the Reverend Jesse Jackson, the Reverend Al Sharpton (an MSNBC host), etc. You'll find as much Christian godliness among liberal American blacks as you will among conservative Evangelical whites. Could you possibly be unaware of that?

45% of Americans Say U.S. Should Be a ‘Christian Nation’ — Art48

A goodly percentage of those are black liberal Democrats. You are misinformed if you think religiosity in America is confined to white conservatives.

Anti-science on the left. Before covid, the anti-vaxxers were upscale liberals in places like Marin county, north of San Francisco. This was commonly known and reported on at the time.

From 2015: "... the anti-vaccination movement is fueled by an over-privileged group of rich people grouped together who swear they won't put any chemicals in their kids (food or vaccines or whatever else), either because it's trendy to be all-natural or they don't understand or accept the science of vaccinations. "

https://www.washingtonpost.com/news/wonk/wp/2015/01/22/vaccine-deniers-stick-together-and-now-theyre-ruining-things-for-everyone/

From 2017: "Because the outbreak started in the wealthy, liberal enclave of Marin County, California, and because some of the best-known “anti-vaxxers” are Hollywood actors, some right-leaning media outlets connected opposition to vaccination to liberals and related it to other “anti-science” beliefs like fear of GMOs, use of alternative medicine, and even astrology. "

https://theconversation.com/anti-vaccination-beliefs-dont-follow-the-usual-political-polarization-81001

Is 'Thank God for dead soldiers' protected speech? — Art48

You'd more likely hear that from a leftist. At least ten or fifteen years ago, before the left embraced the national security state.

In any event, the First Amendment gives everyone the right to say that. Are you making an anti- free speech argument? I didn't follow your point. Of course it's protected speech. The US has extremely strong protections for free speech.

Witch Trials & witchcraft — Art48

Witch trials? Have you got a link to such an event in the US recently? I can't recall such an incident.

As for witchcraft itself, it's almost completely a leftist pursuit. I'm sure you know that.

Witchcraft is the perfect religion for liberal millennials

You know if you come upon a group of people dancing naked in the woods reciting pagan incantations, they're almost certainly on the political left. Not that there's anything wrong with that.

I asked Google, "What does sarte say about beasts?" and their AI gave me a nice answer.

Jean-Paul Sartre's views on animals include:

Consciousness
Sartre believed that animals that can register our presence are conscious, while others are not. For example, clams do not seem to register our presence, and we don't have a strong sense of obligation to them.

Metaphorical use
Sartre sometimes used animals metaphorically to clarify a point in his thesis.

Indifference
Sartre was indifferent to animals, rather than hostile. He was not interested in animals themselves, nor in the moral issues surrounding how we treat them.

Authenticity
Sartre believed that authenticity is central to his moral preconceptions. He believed that some people are more "crab-like" than others, and that this is the opposite of authenticity. — GoogleAI

I liked that last part. Some people are more crab-like. Lacking in consciousness, not worth our regard.

Now this is funny. A discussion of Sarte's view of animals called, "Hell is Otter People." Perhaps you'll find some clues in here. pdf link.

Hell Is Otter People: Locating Animals In Sartre’s Ontology

You've made your point. Don't rub it in. — jgill

Not my intention at all.

For a moment I was thinking q^2<2 normally is q<sqr(2) for positive q, but if irrationals do not exist this inequality is invalid. — jgill

The entire point of $q^2 \lt 2$ is to define that set without reference to irrationals.

In fact that set is the standard example of a set of rationals that's bounded above but has no least upper bound, showing that the rationals lack the least upper bound property.

Twelve pages and I do not pretend to be able to follow all the math. A succinct report would be nice from anyone inclined to provide. Has it been established that the existence of the continuum is strictly a matter of definition — tim wood

Correct, the real numbers are defined as the continuum. They can be proven to exist within set theory, but that has no bearing on what's true in the real world.

The OP's vague ideas about the real numbers were falsified and clarified early on.