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When platonists argue that mathematical objects are real, what exactly do they even mean by them being real? Usually, when we talk about real things, there is a point in space and time where they exist. Consciousness of course may be an exception, but even with that, we have direct experience of it, and know what it is like to be conscious.

What does it then mean for mathematical objects to exist, especially in a different realm?

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  • Existence pf abstract objects, i.e. of "real" objects that do not live in space-time.
    – Mauro ALLEGRANZA
    Commented Nov 25 at 12:21
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    Landry E., Plato was not a mathematical Platonist, Cambridge 2023 (Open access book) - just to dispel some misconceptions.
    – sand1
    Commented Nov 25 at 21:26
  • @sand1 incidentally, OP avoids this by using platonism with a small p, and not Platonism with a big P, as Landry likes to put it. But this is a good point anyway
    – emesupap
    Commented Nov 25 at 22:45

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Translated to mundane speech Platonism means that certain concepts like mathematical objects exists independently from human thinking. These concepts can be discovered by humans, but they are not created by humans.

Plato illustrates this view on abstract objects in his work "Symposion" when Socrates’ retells a speech of Diotima, see Symposion, 210a2ff

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  • This makes sense!
    – Syed
    Commented Nov 25 at 21:31
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Historically, mathematical Platonists may have had an idea of mathematical entities being present in space, but today (after general relativity, quantum mechanics, etc.) it would be difficult to maintain such a posture. Today (some) Platonists postulate a mind-independent realm of mathematical entities (which they tend to describe as mathematical objects). Some are more cautious. For example, Bernays phrased a moderate Platonist position in terms of the objectivity of mathematical "objects":

"there is no fundamental obstacle to attributing objectivity sui generis to mathematical objects."

This is in his article

Bernays, Paul. Zum Symposium "uber die Grundlagen der Mathematik. Dialectica 25, no. 3/4, Conclusions et r'eflexions finales du Symposium permanent de math'ematiques (1971), 171--195.

This is somewhat surprising, since Bernays collaborated with Hilbert on a Formalist programme (perhaps somebody here can explain this comment by Bernays).

Non-platonists reject the existence of either such a mind-independent realm or the objectivity, at least when it comes to infinitary mathematical entities.

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    Over time Bernays came to recognize the limitations of a purely formalist philosophy especially in light of Godel's incompleteness theorems undermining Hilbert's formalism to become sui generis as the nature of mathy truth could be independent from formal systems. Therefore his objectivity sui generis reflects this middle ground in terms of structure not necessarily Platonic realm. Or viewed alternatively, if without objectivity, how different mathematicians can share the exactly same mental concept of certain mathematical propositions in their inherently idiosyncratic subjective way?...
    – Double Knot
    Commented Nov 25 at 20:19
  • @DoubleKnot, mathematicians in fact do not "exactly the same mental concept" as you put it. For example, Leibniz had very different intuitions about infinity than we do in the post-Cantor-Weierstrass era. Leibniz rejected infinite wholes as contradictory; modern mathematicians think they are indispensable. Another example would be the mental concept of an entity like R. Mathematicians traned in the Weierstrassian paradigm have a very different mental concept of R than do those working in axiomatic nonstandard analysis. For example, R_Weierstrass is believed to be infinitesimal-free. ...
    – Mikhail Katz
    Commented Nov 27 at 7:55
  • Meanwhile, R_Hrbacek (or R_Nelson) includes infinitesimals. @DoubleKnot
    – Mikhail Katz
    Commented Nov 27 at 7:55
  • @DoubleKnot, I am not sure I can follow either your comment or the linked post. At any rate, what I tried to point out is that your assumption (that different mathematicians necessarily share exactly the same mental concept of a given mathematical entity) can be challenged.
    – Mikhail Katz
    Commented Nov 27 at 8:17
  • If they share nonstandard models exactly then there won't be IST vs SPOT, there'll be a uni-math foundation for all competant mathematicians, at least in a same era. Everyone has their own subtle preference or emphasis regarding the shape or constitution of their intended model(s)...
    – Double Knot
    Commented Nov 27 at 8:18

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