If the platonic realists are right, the name of that junkyard is the Platonic realm of forms. — fdrake
There's nothing like what there would be if all the mathematical forms instantiated in the same way. — fdrake
But how mathematics looks as a category theorist is quite a lot different from how it looks like under the aspect of set theory. Say, to a category theorist, natural numbers don't look like the names of individual objects, they look like isomorphism classes of sets. Set theory was built out of intuitions about composite objects of multiple elements, category theory was built from intuitions of transformation and symmetry. — fdrake
I have no idea how you took the main thrust of my post to be about beauty or truth. The main thrust is simply that most mathematical objects aren't worthy of study, and agglomerating them all together; producing the final book and the final theorem, far from the ideal vision or ultimate goal of mathematics - produces a writhing mass of irrelevant chaos. — fdrake
The idea that the mathematics that we find valuable forms a Platonic world fully independent from us is like the idea of an Entity that created the heavens and the earth, and happens to very much resemble my grandfather.
Whitehead, the distinguished English mathematician and philosopher, once said that "western philosophy is a series of footnotes to Plato." In so speaking of Plato, Whitehead was putting into memorable words the prevailing opinion of his professional colleagues. But times have changed and recent years [c 1927] have witnessed a powerful reaction against Plato; in the minds of a good many philosophers, reverence has been replaced by execration. Plato is now being seen as playing the role of the villain in the drama of philosophy, so much so that in some circles, merely to characterize a doctrine as Platonizing is to damn it.
Say we take a Platonic stance about math: in some appropriate sense, the mathematical world M exists. The expressions “to exist”, “to be real” and similar can have a variety of meanings and usages, and this is a big part of the issue, if not the main one. But for the sake of the present argument I do not need to define them—nor, for that matter, platonism—precisely.
Recently I happened to be talking to a prominent California geologist, and she told me: "When I first went into geology, we all thought that in science you create a solid layer of findings, through experiment and careful investigation, and then you add a second layer, like a second layer of bricks, all very carefully, and so on. Occasionally some adventurous scientist stacks the bricks up in towers, and these towers turn out to be insubstantial and they get torn down, and you proceed again with the careful layers. But we now realise that the very first layers aren't even resting on solid ground. They are balanced on bubbles, on concepts that are full of air, and those bubbles are being burst today, one after the other."
The fact that the distinction between the expressions 'to be real' and 'to exist' are brushed off says a lot. — Wayfarer
The problem for Platonists is that they have failed to, and apparently cannot, explain in what sense the purported Platonic objects exist, or are real, in some way other... — Janus
Mathematical platonism has considerable philosophical significance. If the view is true, it will put great pressure on the physicalist idea that reality is exhausted by the physical. For platonism entails that reality extends far beyond the physical world and includes objects which aren’t part of the causal and spatiotemporal order studied by the physical sciences. Mathematical platonism, if true, will also put great pressure on many naturalistic theories of knowledge. For there is little doubt that we possess mathematical knowledge. The truth of mathematical platonism would therefore establish that we have knowledge of abstract (and thus causally inefficacious) objects.
Why do you think it might be necessary to state such an argument? — Wayfarer
Rovelli begins with a simple definition of Mathematical Platonism, which "is the view that mathematical reality exists by itself, independently from our own intellectual activities." Now, he asks that we imagine a world M, which contains every possible mathematical object that could ever exist, even in principle. Not only does M include every mathematical object we have currently discovered (integers, Lie Groups, game theory, etc) it also includes every mathematical object we could possibly discover. M is the Platonic world of math. The problem, though, is that this world is essentially full of junk. The vast majority of it is simply useless, and of no interest to anyone whatsoever. — StreetlightX
We don't just not care about them for reasons of utility, we don't care about them because we have a standard of intelligibility which automatically excludes them from our mathematical discourse. — fdrake
So perhaps 'mathematical world, M', is really just a metaphorical depiction of the Platonist intuition of the nature of numbers. But then, it is 'the existence of M' that is thrown into doubt. But maybe this doesn't do anything more than show that this particular way of allegorising Platonism is what is at fault. — Wayfarer
I think that's a point Plato would readily have acknowledged; and a reason why Plato may never have been a Platonist in the modern sense of the term. Modern mathematical Platonism likely is a distortion of Plato's thought, which distortion arose from taking his metaphors literally and misconstruing acts of the intellect -- themselves always portrayed by Plato as outcomes of strenuous and protracted dialectical effort -- as passive acts of contemplation of an independently constituted domain. — Pierre-Normand
Right, but I'm asking if there is a human junkyard of abandoned math, whether constructed or discovered. Because the argument turns on most of math being a junkyard. I'm asking whether this is a hypothetical, or actually historical. — Marchesk
The problem for Platonists is that they have failed to, and apparently cannot, explain in what sense the purported Platonic objects exist, or are real, in some way other than the familiar concrete (human mind independent) existence or reality of sense objects (or at least of whatever gives rise to them), and the familiar ideal (human mind dependent) existence or reality of the contents of thought, emotion and perceptual experience, — Janus
Yes these kinds of objections to Platonism occurred quite early on in the form of questions such as "But is there a perfect form of the turd, or the pile of vomit?". — Janus
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