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
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
Who is doing these terrible things with an anodyne statement like, "Math is what mathematicians do?" And what are they doing?" — fishfry
It's not clear to me whether you're suggesting that my remarks were not pertinent. — TonesInDeepFreeze
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
I'm always amused by this common philosophical example, since Venus isn't a star at all. — fishfry
Isomorphisms have everything to do with structuralism. An isomorphism says that two things are the same that are manifestly not the same. That's structuralism. — fishfry
Medicine is what medical doctors do, thus making it a principally discursive phenomenon. About words rather than bodies. — fdrake
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
If you've not interacted with these people I envy you. — fdrake
I don't think you are making your point. — fishfry
it is the attitude your comment resembles — fdrake
And I don't deny the objective component of science. iow could anyone? — fishfry
I resemble that remark? — fishfry
You don't. If I read your remark out of context, and didn't know your post history, I could read it that way. But I know you didn't mean it like that. — fdrake
Do you mean cultural relativists, postmodernists, etc.? People who think that objectivity and merit are tools of the cis white patriarchy? — fishfry
They're generally working in paradigms like "subtle realism", "new materialism" or the less nebulous actor network theory. — fdrake
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
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
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
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
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
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
I know that sounds weird. Maybe an analogy will help.
So, for example, you know the duckrabbit? — Pneumenon
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
What if math is kind of like that? — Pneumenon
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
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
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
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
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
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
[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
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
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
Maybe I don't need to reduce math to one object after all... — Pneumenon
I hope I didn't wander off too far. And I hope that this is, at least, interesting. — Pneumenon
Yet today, set theory is a basic part of the undergraduate math major curriculum — fishfry
Math is very much a social process. — fishfry
Isomorphisms have everything to do with structuralism. An isomorphism says that two things are the same that are manifestly not the same. That's structuralism — fishfry
And now you react strongly. I wish you'd tell me what you mean. — fishfry
But they do have an effect. Well, It's not like the Catholic Church going against Galileo Galilei and others (or what happened to scientific studies in Islamic societies, that had no renaissance), but distantly it resembles it.I am wondering who these people are that you and ssu think I'm giving comfort to.
Do you mean cultural relativists, postmodernists, etc.? People who think that objectivity and merit are tools of the cis white patriarchy?
If so, I oppose these people. But they're not waiting for the likes of me to give them encouragement. — fishfry
I'm big on scientific objectivity. — fishfry
Good luck finding anyone here that doesn't share your views.But I'm opposed to scientism, and the use of the NAME of science to enforce political, anti-scientific orthodoxy. — fishfry
Thus the make up isn't at all close to the natural 50/50 divide, hence mathematics is male dominated. — ssu
wouldn't this mean then that mathematics is different from being just a social construct of our time? — ssu
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
Good luck finding anyone here that doesn't share your views. — ssu
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
One thing is to keep politics out of things like mathematics. Sounds totally obvious, but we live in strange times... — ssu
Hopefully this clarified my position. — ssu
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 might comment that whereas isomorphisms are very important in mathematics, not all practitioners are heavily involved with them. — jgill
(ground troops, not aviators) — jgill
Reminds me of Tim Gowers's distinction between problem solvers and theory builders. — fishfry
Somewhat similar, but not quite the same thing. I've never particularly enjoyed solving problems, but rather exploring where certain specific ideas in classical analysis lead. The celestial aviators can cruise the heavens taking us ground troops on ethereal adventures. — jgill
We can all believe this. And since people are mathematicians, they can understand the effect when the dean or the higher ups in the universtity or research establishment simply demand that "there should be more women". The obvious reason is simply viewed as toxic. That women have babies and do still become housewives. My wife wrote and finished her PhD when she was nurturing our first baby. When we had second child, she decided to stay home. My income made it possible.Here's a personal anecdote that may be telling: My PhD class had several women. One dropped out for health reasons, and another was the top student, by far. Shortly after graduation she married a forest ranger and became a housewife.
In the international research clique I joined there were several women, but more men than women. A fairly close colleague, a European woman who had left behind a role as housewife, became the holder of an endowed chair at a major Scandinavian university. — jgill
Confused really why you would be in "literal shock" and why talk of having pushback.I said math is what mathematicians do. I stand by the remark. I reiterate my literal shock that this anodyne and obvious statement generated pushback from two people. — fishfry
Good that you used the word "interpreted". It's crucial here.But "mathematics" can be interpreted as the use of data for political purposes. Mathematics is highly political. The NSA employs more number theorists than academia does. — fishfry
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.I used to think so. I probably still do. Still ... Newton and Kant's absolute space and time are reflections of the European paradigm of society in their day. Some philosophers have so argued.. I'm not prepared to go into that too deeply. — fishfry
Sounds interesting. Life in the complex plane. By the way have you seen much of the modern graphing software that's so good at representing complex functions and Riemann surfaces and the like? Don't you wish you'd had that back in the day? I wish they'd had LaTeX, I always had bad writing. — fishfry
Retired for 24 years. Lots of things slip by. Hard enough to persist along the lines of mathematical thought I know about. — jgill
Get involved in philosophical discussions about knowledge, truth, language, consciousness, science, politics, religion, logic and mathematics, art, history, and lots more. No ads, no clutter, and very little agreement — just fascinating conversations.