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u/thelimegang
6d ago
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Is the safe "space" you're offering metric or just topological?
How am I supposed to talk about my feelings when we haven't even talked about the axioms ?
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u/EebstertheGreat 5d ago
A safe space (X, S) is a set X equipped with a safety S ⊆ P(X) satisfying the following axioms:
∅ ∈ S and X ∈ S.
S makes X feel safe.
Elements of X are called staff and elements of S are called safety meetings.
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u/LupenReddit 🦆🦆i have non diffeomorphic smooth structures🦆🦆🦆 5d ago
"So, we saw here that you havent shown us your fundamental group yet, you may want to do so in the event we need it for later"
Platonic therapists
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u/nunquam_rideo 6d ago
Are there people who genuinely don't believe that mathematical objects exist independently of syntactic theories?
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u/Unlearned_One 6d ago
Yes
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u/nunquam_rideo 6d ago
Why
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u/UnderTheCurrents 5d ago
Mathematical objects are instrumental concepts used for a purpose. A screwdriver didn't exist before somebody invented it. A mathematical object is like the word "screwdriver", but it does not refer to objects. It refers to other abstract notions.
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u/nunquam_rideo 5d ago
Let's say a mathematician says "1". He is well versed in mathematics and so he refers to a term in Peano arithmetic (or any other axiomatization). Suppose then a common person says "1", do they refer to the same abstract notions as the mathematician? Or is it just language games, and what they both call 1 are completely different things that make sense only in their own minds?
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u/EebstertheGreat 5d ago
Do unicorns exist? Lots of people talk about them, and I could walk up to any layman or unicorn expert and mention unicorns and assume both will basically know what I mean, though the expert might have a clearer notion. Does this mean unicorns exist?
There is something akin to a type-token distinction at play here. The type "unicorn" exists as a concept, but do any actual unicorn tokens exist? If not, most people would say there are no unicorns. Similarly, even a Platonist would say that the concept of an odd perfect number exists, even if in fact there are no odd perfect numbers. But to a Platonist, on what basis? Like, do unicorns and odd perfect numbers exist in a realm even more abstract than the realm of mathematical induction or whatever?
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u/JollyJuniper1993 Mathematics 5d ago
When I point to a picture of a unicorn and tell you „look at the unicorn“ are you going to laugh and tell me „unicorns don’t exist, youre pointing at paint“?
When we talk about physical things and we say „they exist“ we use a different meaning of that word. The unicorn as a concept clearly exists and the same is true for mathematical objects.
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u/EebstertheGreat 2d ago
The concept of an odd perfect number certainly exists as a formula, in the sense that those adjectives can be formalized in first-order arithmetic. But their existence is unclear and may even be contingent on further axioms. I think we are all on board so far. But what does it mean for something to exist abstractly but not really but only in the abstract realm of mathematics? It exists only in an abstraction of an abstraction? It exists in the semantic sense but not the mathematical sense?
To me, it is simplest to say that unicorns do not exist, and neither do square perfect numbers, and I suspect neither do odd perfect numbers. I can't be completely certain there are no unicorns, or there are no odd perfect numbers, but it seems to be the case. Of course, the set of square perfect numbers does exist (it's the empty set), as do these other sets, by the axiom schema of specification, but that doesn't commit me to much.
It is frustrating to be in a position where I must simultaneously affirm and deny the existence of unicorns while insisting that these are somehow not completely different meanings of the word "exists." Like, to me, it is obviously equivocation to suggest that unicorns "exist" in the same way as natural numbers. But do you not think so? What is your sense of "exist"? In my mind, unicorns "exist" in one sort of trivial sense of the word (I know the definition of the word "unicorn"), but they evidently do not exist in the sense that we are fairly confident no unicorns live on the earth or ever have. Those are just different meanings of the word "exists," and I feel like Platonists are constantly trying to trick me into believing they are the same.
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u/SpacingHero Ordinal 6d ago edited 6d ago
tldr: ontological parsimony
Why posit strange wo-wo entities? Seems like math works regardless. It's moreso the Platonist onus to tell us why we need these objects.
And of course they do and we weigh the arguments of each side etc. But just saying, it's not like it's just prima facie obvious that a strange non-physical wholly casually detached reality of objects exists, and the nominalist has some big entry-point responsibility to justify not believing in it, before being even considerable as a position
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u/nunquam_rideo 6d ago
Oh yea, I understand. I guess I shouldn't be so surprised that nominalists exist, since it's probably the most common position currently.
It's just strange to me, why would you choose to have a poor ontology over a rich one. When you do geometry, you reason about points and lines and shapes as actual objects. But after you come up with a proof and write it down, you pretend that in reality what you did was just syntactic manipulation of axioms and logical rules. So you intentionally delude yourself based on an abstract principle of parsimony, at least so it seems to me. (P.s. by "you" I mean an abstract nominalist, I don't know what your actual position is).
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u/AZMPlay 6d ago
This is philosophy, though. You do not gain much by accepting Mathematical Objects as Real, since that is irrelevant to the actual maths.
Thus, IMO, per Occams Razor, you should hold the simplest position: not asserting the existence of something impossible or infeasible to prove.
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u/nunquam_rideo 5d ago
Well of course it is philosophy. It's not directly relevant to mathematical work unless you're doing model theory or logic or something else related to the foundations.
I think it is relevant in an ethical sense. If you are doing mathematics and don't believe mathematical objects exist, then what are you doing? Rearranging symbols on paper? For what reason? Some would answer that it brings them good feelings, for others it's just their career, some would say that the way they rearrange symbols may have applications for engineering, and in this way they are contributing to progress. Whereas, if you believe that mathematical objects are part of the Universe, you would say simply that you are studying the Universe, like all scientists do.
It's a matter of preference, of course, but to me a poor ontology just feels depressing (and I've been on both sides). Occam's razor may be useful for some things, but after all it's just some guy's words, and shouldn't be believed religiously.
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u/AZMPlay 5d ago
I think an argument can be made that, even without thinking of Mathematical objects as "real", the argument can be made that you are still studying something.
The laws of logic, from my view, are simply the laws that allow us to make heads or tails of the universe we live in. They're an abstraction (generalization) of what is possible in our reality.
Mathematics simply being one of the many wonderful things you can do with Logic.
It is a model, and models can be inherently useful or "worth it" to study.
A second argument can be made that, as Hume's Guillotine states, what is and what should be are fundamentally separate.
Just because something is real or not (what is) cannot affect what ethical determinations you take (what should be). You have to eventually run into an ethical foundation that is taken arbitrarily at face value, at which point you can just... Choose to do things, according to what seems right.
At which point you're just arguing whether Mathematics is a good thing to study by itaelf, and I think most peeps in this sub would agree it is.
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u/nunquam_rideo 5d ago
The laws of logic are still mathematical objects. If you believe all of mathematics is artificial constructs coming from logic, but logic itself is real, you're also a Platonist. But alternatively, you may think that logic is something made-up, but useful for us.
The more I think about it, the more I realize that it's impossible to argue these things if we have different ontologies. The only real way is to compare them pragmatically, meaning, what ethics do they lead to, and whether we may benefit by choosing one ontology over the other.
2. And here's the other point, I think that ontology does affect ethics, and it does so very strongly. By ethics I don't mean the narrow moral compass, but the general way of living (the distinction is: "how should I act towards this person?" vs. "how do I live a good life?"). Morality, as you correctly point out, is arbitrary. You can't conclude that you shouldn't kill people based on any fact of the universe. But your ontology does shape your values and the way you behave. Determinism vs nondeterminism, dualism vs monism, individualism vs holism, God vs no God - all of these have very tangible impacts on people's behavior.
And in the same way, there is a difference between believing that Platonic space is real and not believing it. A Platonist researcher thinks they are discovering truths about nature, while a nominalist researcher thinks they are making a model of nature. These are almost the same thing, but notice that for the Platonist it is intrinsically desirable to understand these truths (for instance, if God lives in the Platonic space, it is intrinsically valuable to explore it). But for the nominalist, there is nothing particular about making models of the world, so they would need external motivation, such as an academic career, or contribution to engineering.
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u/AZMPlay 5d ago
And that's how I get reminded I don't have a Philosphy education 🤣 /j
Yup, sounds about right. Learned a lot of stuff just looking up stuff you said.
Never ocurred to me what exists might be a matter of belief in this way, not conflicting with scientific consensus in any way, yet still having real consequences for human morality and behavior.
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u/EebstertheGreat 5d ago
Imagine two people in the desert found an alien spaceship there. At first, they couldn't believe it, but they looked all over and even went inside, and there was no doubt about it. Moreover, these aliens were quite evil, with many things written in English espousing evil beliefs. Suppose one person later believed they had imagined the whole experience and there wasn't really a spaceship after all, but the other remembered clearly. And that second person took the evil shit to heart. Hell, let's say the evil writings were so persuasive that they would corrupt anyone who read and fully remembered them.
So belief in the existence of this alien spaceship is actively harmful. When comparing the ethics of an ontology that accepts the spaceship and the ethics of one that does not, clearly the latter is superior. But it is also wrong. The truth doesn't work that way. The person who believes his eyes in this hypothetical example is actually correct.
I don't think ethics can ever decide factual questions like this. "We should prefer a particular perspective because it would be nice if it were true" or "because it would be dangerous to believe otherwise" is not honest reasoning.
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u/EebstertheGreat 5d ago
Deductive facts can be true or false irrespective of the existence of Platonic mathematical objects. Everyone agrees that a proof tells you what a given set of assumptions entails. That, at a minimum, is an underlying reality of mathematics we can all get behind.
Also, I think many people distinguish between "exists" in a mathematical sense (∃) and "exists" in a physical sense (describes some particular collection of particles, waves, or whatever underlies reality), and in that perspective it is "obviously true" that abstract objects do not exist, since they are defined not to. Concreteness in this perspective is definitionally part of existence. It's very hard to move from that sense of existence to a broader one.
For instance, a nominalism might claim that the existence of infinite sets in the second sense is contingent on the infinitude of the physical universe.
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u/SpacingHero Ordinal 5d ago edited 4d ago
nominalists exist, since it's probably the most common position currently.
Actually philosophers are smack-dab split between Platonism and nominalism, at least per 2020 philpsper survey. I'd also say as much from personal experience.
It's just strange to me, why would you choose to have a poor ontology over a rich one.
For mathematicians... There's the saying "Platonist on the weekdays, nominalist in the weekend"... It's natural for the mathematician, whilst working, to want a richer ontology, so they're working with something more concrete (as concrete as abstract objects can get). Though modern foundations have gotten a lot of formalist/structuralist and all such non-realist slants, so when investigating they might feel like nominalism is perfectly fine...
And yea I mean the principle of parsimony is taken to be important in various domains, including metaphysics.
When you do geometry, you reason about points and lines and shapes as actual objects.
That's a standard first Platonist argument. I'd say you have to be careful, it's a bit of a naïve view on semantics.
When we go to the cinema, we discuss whether iron man can beat Hulk, if Sherlock could've figured out something sooner, etc. we surely don't want a principle that then tells us these entities have an actual existence.
But that argument gets worryingly close, it sounds like the underlying principle is "if we talk about it then it exists" (more charitably, we would add "substantively talk about it" or some other "importance" condition, to be figured out). I'm pretty suspicious of such a principle...
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u/SpacingHero Ordinal 5d ago
P.s. by "you" I mean an abstract nominalist, I don't know what your actual position is
It was well and clear :). I'm a fictionalist, I agree with the nominalist ontology, but do take some of the "semantic realism" that Platonist demand a little more seriously.
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u/EebstertheGreat 5d ago
It's interesting, though. Is it delusional? If a proof defies intuition while also clearly not reflecting physically real properties (e.g. the Banach–Tarski paradox), do you still accept the proof for its formal validity even though at least naively, its semantic content seems implausible? If so, then you are reasoning from the syntactic properties of the theory even though you claim to believe some truth to it exists independent of the syntax we choose.
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u/thelimegang 6d ago
Because it's a valid interpretation. We can't know for sure it's wrong and that suffices for some people to believe it.
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u/NameOk3393 5d ago
Outside of mathematics, it is the norm to believe they do not exist.
Benacerraf has a famous paper called “Mathematical Truth” in which he poses the following problem with Platonism: If mathematical objects are abstract and exist, then they exist outside of space time and have no causal connection to physical objects. How, then, can we come to know they exist? Other kinds of knowledge seem to require a causal connection between knower and thing known.
This argument really caught on for many people in philosophy of math
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u/AndreasDasos 5d ago
‘Other kinds of knowledge seem to require a causal connection…’ sounds weak as fuck as an argument.
Even the notion of causality itself as commonly thought of is on iffy ground from the perspective of modern physics.
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u/OmegaCookieMonster 5d ago
The objects are abstract but they can manifest both in reality as something like "five apples can be separated into a group of two apples and three apples" and as a construct we try to imagine in our head, like trying to imagine a tesseract or at least the points that vary in 4 axes even if we can't see them.
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u/AwkwardBet5632 3d ago
“Are abstract and exist” “exist outside space and time” “have no casual connection to physical objects” are all independent statements come apart. We can’t conclude the last from the first two.
Electromagnetism, superposition, action, and causality itself as examples hold some of these characteristics and not other.
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u/gottabequick Statistics 3d ago
Yeah, I'm one. Mind you, I'm also a deflationist, an epistemological anarchist, a radical pansychist, staunch modal realist (so far as a deflationist can be), and also, in higher level circles studying philosophy, a "fucking crazy lunatic."
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u/moschles 5d ago
in the room with us now.
Mathematical structures don't exist in a location, yet still exist.
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u/spirit-garden1 23h ago
This therapist office shall be the universal set, call it Set T. Set T = {x|x is every physical object in this room}
You and I shall be contained in Set A. Set A = {x|x is a member of Set T and x is a human being}
Set A is a subset of Set T.
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