Hard Core Platonists

You are one if: If you are one: keep faith! I was one too, and I got cured. -- StephanHouben

An accessible introduction to the controversy from a platonist point of view: http://research.microsoft.com/~gurevich/Opera/123.pdf

Nnother accessible introduction from a constructivist point of view: http://www.math.fau.edu/Richman/docs/intrview.pdf

But what about all the others? Let's do a vote! We all know the rules now from WikiGreatFoobarLists. We'll list some mathematicians, mathematical physicists, philosophers, linguists and scientists below who knew or maybe know a bit about this stuff, and rate them from 0-9 as follows:

The big list (please interfere with) with the latest WikiBallotBoxes:

It is certainly pleasant to see one's name in the above list. -- StephanHouben

I was taught by a Platonist in my "Intro to Philosophy" course. He claimed that he was the only one left in the department, and would get mail addressed to "Platonist, UF Philosophy Dept." -- Pete Hardie

AlbertEinstein as a Platonist? The same guy who rejected the concept of "objective time" in physics because it could never be measured? Einstein was a romantic, a serious atheist who made conspicuous references to "God", that doesn't mean he was a Platonist.

Einstein was a physicist. I'm not sure on what basis the scores given him were made - certainly, nothing I've read about him shows him to be in one camp or another. Clearly, he thought that the field equations described something real, but I think that platonism, as in believing that integers exist, is different from physicists believing their equations describe the real world or is only a mathematical approximation to it. So I'd remove Feynman from the list also. You did notice that the only person to rate him put him at 2, yes? This is a list of people whose views might be interesting, not a list of hard-core platonists. It seems dangerous to be commingling physicists and mathematicians in this enquiry. Or rather, to be commingling physical thought and mathematical thought. Platonic issues, I think, play very different roles in these two fields.

I'd add Kronecker, who is known to have said:
  God invented the integers, all else is the work of man.

So integers are Platonic and everything else is constructed? Perhaps this poll is not so useful? Or is this what is meant by "soft-core Platonism"?

Why does one need to reject the Continuum Hypothesis (which is what #2 above amounts to) in order to be a HardCorePlatonist?? Never encountered that before.

#2 is nothing to do with the Continuum Hypothesis. CH says that there are no cardinals between aleph-0 and P(aleph-0). #2 above is about ordinals, not cardinals, and more specifically about the fact that the class of all ordinals, even though it's "obviously" well-ordered, has no corresponding ordinal. Stephan's point is that if you are bothered by this, it suggests that you think the ordinals are really "out there".

The two paragraphs above are broken. #2 (if we count from 1, anyway) is about the Continuum Hypothesis, but doesn't say you have to reject it to be an HCP. It says that if you think it's a real question then you're an HCP.

The top list seems a little bit odd...thinking the natural numbers really exist somewhere (nowhere? everywhere?) is clearly Platonism, but what about the others? I'm not really sure about any of this, but here are my thoughts - please CutThemToRibbons? if they are erring:

...whether there is an infinite, uncountable set of numbers not equivalent to the reals?

The very fact that the proposition is undecidable means that there isn't one - not in ZF. It's just that ZF can't prove that, so you could construct a bigger system which has such a thing and is still consistent, just like you can add SuperNaturalNumbers? to TNT (typographical number theory).

ZF isn't a model, it's a theory. It doesn't answer all the questions. It's an error to say "there isn't one - not in ZF". It might be the case (though, as it happens, it's not) that there is no model of ZF that lacks all the things that ZF can't prove the existence of.

The point is that there is a model for ZF satisfying ZF+CH, so the extra sets aren't necessary. I know the distinction between model and theory but how relevant is it here? When you talk about the Peano axioms there is a certain system that readily comes to mind, and it doesn't include Supernaturals although they are not explicitly forbidden.

Theories (of set theory) do not contain sets. ZF doesn't contain the empty set, or the set of all sets, or anything like that. The Peano axioms do not contain the number 0. It's clear that you have a preference for small models, but the theory itself doesn't have.

It's probably true that we understand the "standard model" of N as clearly as we understand the Peano axioms, and that it's reasonable to prefer that model to others. I don't think anything comparable is true for set theory. The main thing we know about our intuition, ever since Russell, is that it's broken. :-) So you're welcome to think that you have a "standard model" of ZF in your head, and that it definitely satisfies CH; but I think you have less in your head than you think you have.

For instance, does your model of ZF satisfy the Axiom of Choice? (There are models of ZF that do, and there are models that don't.) If it satisfies AC, then (for instance) it contains non-measurable sets. I'd guess that your preference for minimality makes you prefer not to satisfy AC. Does it satisfy the Generalized Continuum Hypothesis? Since you prefer CH, I conjecture that you prefer GCH too. Well, if you have the preferences I think you have, then you're out of luck, because actually GCH implies AC. :-)

Does it indeed? For some reason I remembered the implication as going the other way...a while ago I looked for an example of two contradictory minimalist assumptions, and because of this misunderstanding didn't find one. So OK, that changes everything, and thanks very much for the info! -- JG

I should admit that I'm being a little cheeky here. There is a sense in which ZF really does have a "minimal model", and as it happens that model satisfies AC. I'm just pointing out that waving your hands and saying "ZF doesn't prove this thing exists, therefore this thing doesn't exist in ZF" is liable to get you into a terrible mess.

...but then it that case, how does ZF still have a minimal model?
...BigOmega is not an OrdinalNumber?...

The SetOfAllSets isn't a ZF-set, either, but it seems that you can have it in a weaker axiom system. I'm not totally confident in that, but I suggested it on GoedelsIncompletenessTheorem and no-one has had any qualms yet.

I think the impossibility of BigOmega being an ordinal goes deeper than that. By the way, not all set theories with universal sets are "weaker" than ZF.

...you think that reams of paper filled with funny squiggles have some meaning...

First, to kill the obvious - clearly they have some meaning, since we understand certain symbols to be the same regardless of position or font or so forth. Obviously they have meanings the same way words do, by mapping to mental equivalents of the same symbol, and that's not what we're talking about.

I certainly don't know enough to say, but can one say they have no meaning and still explain MathematicalCoincidences? - how nontrivially different axiom schema often tend to give the same results? For instance, the many systems that turn out to be ZF or Turing machines, or the underlying connections between Lie groups and polyhedra or limits of quotients and limits of sums.

...you are trying to reduce all of mathematics to set theory...

Doesn't saying mathematics is squiggles essentially do this for you? Well, maybe not reducing mathematics to set theory, but certainly reducing it to something - one could tell you a list of properties that any string of symbols satisfies, and if that's all math is, poof! - you have reduced it all to string theory (and how do you talk about a formal system anyways, without "sets" of theorems and axioms that satisfy some sort of properties?)

I don't want to be promoting an agenda (and certainly not to get in a holy war), because I honestly don't know about any of this stuff...it would be nice for someone who knows the field to come in and explain all the different viewpoints, and how each of them works or doesn't work. At least, if people are going to be RampantAntiPlatonists?, to know how exactly the alternatives handle things.

-- JoshuaGrosse

I'm led to wonder - is there a special mathematical sense in which the word 'Platonism' is being used on this page? Because as far as I can tell, it has only the loosest connection to Plato's philosophy. -- MossCollum

It's the usual sense restricted somewhat. Remember Plato thought all sorts of abstractions, like chairs and happiness and truth and red, existed somewhere as eternal ideals. A mathematical Platonist would be someone who thinks that mathematical abstractions at least have this sort of existence.

I read an article recently (10/1/2003) on a possible resolution of the Continuum Hypothesis. Start by looking at all the mathematical problems even remotely related to the CH. Now try to find a single axiom that resolves all of them one way or another. Any axiom that does so is called an elegant axiom. It turns out the elegant axioms all say the same thing.

What exactly do the elegant axioms say?

See also: SoftwarePlatonism

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