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  1.  
    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 5th 2020

    Initial writeup.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 5th 2020

    That is a good example!

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeApr 6th 2020

    added toc, and changed page name to singular

    diff, v2, current

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeApr 6th 2020

    The various parts of this page seem inconsistent to me. If a mathematical object is interpreted by a formalist as a term, then it must be a precisely defined thing, so that the nebulous not-precisely-defined notion of (∞,1)(\infty,1)-category would not count as a “mathematical object” even to a Platonist.

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeApr 6th 2020

    Do Platonists actually believe that even imprecisely defined notions like (∞,1)(\infty,1)-category “exist” in the Platonic sense? I have generally interpreted Platonist views of mathematics to be talking about the existence of precisely defined mathematical notions such as quasicategories or complete Segal spaces. (I suppose in order to answer this question one would first have to answer the question of whether there are any actual Platonists not made of straw, which I don’t really know the answer to either.)

    • CommentRowNumber6.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 7th 2020

    Do Platonists actually believe that even imprecisely defined notions like (∞,1)(\infty,1)-category “exist” in the Platonic sense?

    I’m certain most of them do. Before Eilenberg and Mac Lane knew how to construct Eilenberg-Mac Lane spaces, they had an abiding faith that such had to “exist” (and I’m not even claiming that Mac Lane, for one, considered himself a Platonist – certainly he didn’t later in life).

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 7th 2020

    I’m not sure what is meant in this conversation by ’Platonist’. Do people mean an advocate of the currently held position within the philosophy of mathematics known as ’Platonism’? I find it painful to read such unfruitful care being lavished on a topic, but one should probably have a look at SEP: Platonism in the Philosophy of Mathematics.

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeApr 7th 2020

    Before Eilenberg and Mac Lane knew how to construct Eilenberg-Mac Lane spaces, they had an abiding faith that such had to “exist” (and I’m not even claiming that Mac Lane, for one, considered himself a Platonist – certainly he didn’t later in life).

    I don’t consider that an example for two reasons. First, my question was specifically about Platonists, so we would need evidence that Mac Lane did consider himself a Platonist at that time. (Otherwise, the relevant meaning of “exist” is totally different.) Second, the notion of Eilenberg-MacLane space is precisely well-defined, whether or not one knows how to construct them.

    I’m talking about a belief that “model-independent” (∞,1)(\infty,1)-categories really exist right now in a Platonist sense, even if there is no precise mathematical way to define them; which is different from believing that there is a precise definition of such objects that we will discover one day. This belief is how I read the current text on the page; if that wasn’t what it was intended to claim, then maybe we can change it to be more clear.

    Do people mean an advocate of the currently held position within the philosophy of mathematics known as ’Platonism’?

    That’s what I mean. What else might one mean?

    • CommentRowNumber9.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 7th 2020

    What else might one mean?

    Well, the discussion above seemed to be working with some informal mathematicians’ sense (arguably closer to Bernays and Gödel’s, and very likely closer to Plato’s, than the current version). Your question

    Do Platonists actually believe that even imprecisely defined notions like (∞,1)-category “exist” in the Platonic sense?

    wouldn’t make sense to a current platonist. It’s not a question of the existence of “notions” (or concepts or ideas), but of “objects”. If they’d ever heard of them, they might wonder whether (∞,1)-categories exist.

    • CommentRowNumber10.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 7th 2020

    I’m talking about a belief that “model-independent” (∞,1)(\infty,1)-categories really exist right now in a Platonist sense, even if there is no precise mathematical way to define them; which is different from believing that there is a precise definition of such objects that we will discover one day. This belief is how I read the current text on the page; if that wasn’t what it was intended to claim, then maybe we can change it to be more clear.

    Better ask Dmitri directly what he meant then. I think I can see how you come to that reading.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeApr 7th 2020

    Yeah, Platonism is not the idea that any given definition has an instance.

  12. First, my question was specifically about Platonists, so we would need evidence that Mac Lane did consider himself a Platonist at that time.

    I don’t see the relevance of what MacLane defined himself, unless you work with a sociological definition of platonism, which you don’t according to your answer:

    Do people mean an advocate of the currently held position within the philosophy of mathematics known as ’Platonism’?

    That’s what I mean. What else might one mean?

    The point of dispute seems to be the epistemological content of platonism. The “currently held position within the philosophy of mathematics known as ’Platonism’” is void of supposing any “access” (meaning ways to know about) to particular mathematical objects. But for the disputed example regarding (∞,1)(\infty, 1)-categories, a mathematician exactly needs such an (perhaps not very reliable) access to Platonic objects in order to have any confidence in its existence.

  13. Yeah, Platonism is not the idea that any given definition has an instance.

    Just to make sure, what is it that you reject (I guess you mean the second option):

    1. that every definition has its own corresponding object (so even if two definitions are equivalent (in some sense); or perhaps definitions have an instance even if there can not be an object corresponding to them?), so basically there is a correspondence between “instances” and the strings expressing the definitions

    2. objects/instances correspond to the definitions, but two definitions can share the same object.

    • CommentRowNumber14.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 7th 2020

    To clarify what I meant when I wrote the article:

    Newton, Leibniz, and Euler worked with infinitesimals that (in their time) were not precisely defined, but they accepted their existence.

    Likewise, right now we may not have a precise definition of (∞,1)-categories (other than specific models for them), but some of us may accept their existence nevertheless.

    • CommentRowNumber15.
    • CommentAuthorMike Shulman
    • CommentTimeApr 7th 2020

    It’s not a question of the existence of “notions” (or concepts or ideas), but of “objects”.

    Sorry for using a confusing word. I’m not sure exactly what the difference is between saying that a notion exists and saying that an object exists, but I meant whatever one you’re talking about, which I guess is “objects”.

    • CommentRowNumber16.
    • CommentAuthorMike Shulman
    • CommentTimeApr 7th 2020

    some of us may accept their existence nevertheless.

    Are you such a person?

    • CommentRowNumber17.
    • CommentAuthorMike Shulman
    • CommentTimeApr 7th 2020

    And if so, what exactly do you mean by “existence”?

    • CommentRowNumber18.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 8th 2020

    I guess so. I act and interpret them in the same way as I would act and interpret the notion of an electron in physics. While we do not have a precise rigorous mathematical description of an electron that encompasses all of its uses, electrons do cast (Platonic) shadows in the mathematical world, e.g., Schrödinger’s equation, etc.

    So electrons currently exist in the same way as (∞,1)-categories. And perhaps one day we will have a precise mathematical definition of an electron that does encompass all of its uses, just as one day we might have a precise definition of an (∞,1)-category (in homotopy type theory, maybe?).

    • CommentRowNumber19.
    • CommentAuthorMike Shulman
    • CommentTimeApr 10th 2020

    I certainly believe that model-independent (∞,1)(\infty,1)-categories exist as an idea, which is the same sense in which I believe that precisely defined mathematical objects exist. But I would not call the former a “mathematical object”; I would reserve that name for things that have been precisely defined. Perhaps others feel differently; but I think it would be good for this page to at least mention that there is such a dichotomy. What would you think about that?

    • CommentRowNumber20.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 11th 2020

    I am certainly fine with mentioning this. The very point of the page was to point multiple (possibly incompatible) points of view.

    • CommentRowNumber21.
    • CommentAuthorMike Shulman
    • CommentTimeApr 11th 2020

    But, to be clear, do you believe that model-independent (∞,1)(\infty,1)-categories exist in the same sense that, say, quasicategories do?

  22. Added the link mentioned by David to Stanford Encyclopedia of Philosophy.

    diff, v3, current

    • CommentRowNumber23.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 11th 2020

    Re #21: “Exists” is ambiguous here, since it also has a formalist rigorous meaning (∃\exists). This is not the meaning used above, and only quasicategories exist in the formalist sense.

    But (∞,1)-categories and electrons definitely platonically exist in the same way as quasicategories and Schrödinger’s equation.

    • CommentRowNumber24.
    • CommentAuthorNikolajK
    • CommentTimeApr 12th 2020
    • (edited Apr 12th 2020)
    I also think, Dmitri, that people generally have something stronger in mind when they say "mathematical Platonist," than how the word is used in the article (if I extrapolate from the use of the term in that article). And, Mike, indeed you'll find people on the web who adopt such stronger views, e.g. on MO. E.g. take some property P on the natural numbers that's computationally undecidable. You'll find people who assert that each number n "exists" and that for each of them P(n) has a value, even if it's not accessible to "us".

    Dmitri, your "Platonic shadow", if I read you correctly, is in itself a weaker notion. E.g., your electron example sounds like the wonder about the famous "unreasonable effectiveness." If I may try to describe the intuition, the thinking goes like so: Even if we have yet no mathematical description around, which does fully satisfy everybody researching the phenomena associated with the electron idea, we surely observe that e.g. This and not That differential equation turns out to be a fruitful coarse description some of said phenomena. Now, by a logic of causation, we project that there's some entity that indeed is the reason for which model functions as a useful approximation, and in that sense we speak of the existence of "the electron."
    However, I think to draw examples from both physics and math introduces more issues than it does good, since physical realists are much more common. On the math side, e.g. with infinity-categories, if some nice and interesting object is finally found, mathematicians may have an easier time to declare to have reached the bottom of the questions that concerned their previous generations: Mathematicians may come to the consensus that These 15 intuitions were indeed correct and are realized by the nice object, while Those 2 intuitions were just a daze the source of which can now be well understood.
    There's yet a weaker reading of the use of the term, in that any "idea" has a realization in our discourse: Of course, Spiderman doesn't exist, but Spiderman also inspires many people occupies peoples thoughts and activities.
    Even weaker yet, there's always meta-properties of any idea. The number seven is defined as the successor of six, but of course there's more to it: It was also the favorite number of Marilyn Monroe.

    With this lengthly attempt at laying out some notions, let's come to that section:
    >2. Formalist definition
    >From a formalist viewpoint, a mathematical object is a term in whatever system of formal logic we happen to be using at the moment. [...]
    >3. Platonic view
    >Of course, there is more to mathematical objects than their syntactic aspect exhibited above in the formalist definition.
    >A good example is

    On the one extreme, if we read the last sentence in a loose Marilyn Monroe's favorite number kind of way, the statement tautological, and even the formalist wouldn't deny that there's more to it.
    If, however, you read it in a strong Platonic way, then this is quite a claim. As in, "The old Romans prayed to various Gods. Of course, there's only one God."
    In the end, I think the controversy here stems from the fact that Formalists and Platonists are philosophies that both have proponents that are rather convinced their view is a correct one, so when the section says "A: View A. B: Of course, there's more to it! View B.", then this is read as philosophical commitment to the latter.
    • CommentRowNumber25.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 12th 2020

    since physical realists are much more common.

    Which is why I gave this analogy with electrons, to make connections with the larger community of physical realists.

    then this is read as philosophical commitment to the latter.

    I rewrote this sentence to avoid such impressions.

  26. Expanded the idea section pointing at the possible controversies.

    diff, v5, current

  27. Changed section heading from “Platonic view” to “Platonic views”.

    diff, v5, current

    • CommentRowNumber28.
    • CommentAuthorDmitri Pavlov
    • CommentTimeFeb 1st 2021

    Redirect: Platonic form.

    diff, v6, current

1 to 28 of 28

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