# Can a physicalist be also realist about mathematical objects?

Is it possible to believe that mathematical objects enjoy some kind of mind-independent existence while holding physicalism?

And if they are mind-dependent, should one embrace constructivism necessarily?

It seems problematic to reject classical mathematics on the ground of physicalism, since physicalism itself require classical mathematics. Of course it's possible to build up analysis from the constructivist point of view, but scientists usually use classical math. So how to reconcile these claims?

EDIT: Here's the problem

If physicalism is true, mathematical objects to exist should be physical

Mathematical objects are either something implemented in the humand mind (constructivism) or some kind of things that exist in a non physical realm.

Therefore, if physicalism is true, mathematical objects exists only in the human mind (anti-realist stance, constructivism)

Physical statements make use of classical mathematics (which is non constructivist)

Therefore, if physicalism is true, mathematical objects exist not only in the human mind

• Actually, mathematical objects do not have to be physical. "Mathematical objects" today are not what they used to be to Plato, ideal analogs of material objects. Physicalism obviously admits physical laws and properties in addition to material objects. One can believe in reality of bundles of such that function sufficiently like material objects to be called "objects", and even argue that they are indispensable in physics. This is sometimes called thin realism. Commented Jun 27, 2018 at 17:28
• @Conifold Care to expand further? "One can believe in reality of bundles of such that function sufficiently like material objects to be called "objects"" bundles of properties of physical objects? For example: how would the set of natural numbers exist in such a setting? Commented Jun 27, 2018 at 18:46
• Look under the second link, they give many references. Maddy's Realism in Mathematics is probably the standard reference. Thin realism assigns them similar status to that of abstract objects/universals generally, but the link to something material can be much more convoluted than the link of roundness to round objects. Commented Jun 27, 2018 at 21:05
• @Conifold I didn't know about Maddy's Realism. Thank you for the reference, it seems interesting Commented Jun 29, 2018 at 10:56

We need to distinguish believing in the actual existence of mathematical objects, and believing that pretending that the mathematical objects exist will only lead us to true conclusions regarding the objects that actually do exist.

To embrace physicalism and physics, rejecting the former but accepting the latter is the way to go.

As an aside, it may be worth pointing out that most of mathematical physics does seem to be constructive anyway. Even if not, the objection to non-constructive math used in physics should be that without being able to compute the predictions, we cannot empirically test them, rather than an ontological preference for intuitionism over Platonism.

As an aside to the aside: Mathematical economics, on the other hand, is full of non-constructive math...

If you want a precise explanation, you should think on what existing means. I have realized that for something to exist for an observer, it has to be in the mind of the observer. So stuff cannot exist if it is not observed. On the other hand, math is a language we humans use to describe what we see— a formal language if you want, but it still is a language.

• "something to exist for an observer it has to be in the mind of the observer": what about reductio ad abursum proofs? Here we don't really have in mind "nothing": we just conclude (from the principle of excluded middle) that something must exist because the opposite proposition (the non-existence of the object) entails a contradiction, even though we don't have in mind that object. Your position seems to imply constructivism (mathematics as the product of the human mind) Commented Jun 27, 2018 at 14:10
• "stuff cannot exist if it is not observed ": have you ever "observed" Napoleon, or Waterloo battle ? What about whales or orangutans ? Commented Jun 27, 2018 at 14:13
• You still have it but in a different way, to conclude something you need to have information which you gathered by looking something maybe in the past, but something that was absurd in the past does not imply that it keeps being absurd in the present because to deduct you need experience.