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Should we have ontological commitment to mathematical objects? Watch

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    Essentially, should we subscribe to Platonism in the interests of science? To sum up the argument I subscribe to (Quine-Putnam indispensability):

    We should be ontologically committed to the existence of the entities required for the workings of our best scientific theories
    Mathematical objects are required by our best scientific theories
    So we should have an ontological commitment to mathematical objects
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    (Original post by Seelowe)
    Essentially, should we subscribe to Platonism in the interests of science? To sum up the argument I subscribe to (Quine-Putnam indispensability):

    We should be ontologically committed to the existence of the entities required for the workings of our best scientific theories
    Mathematical objects are required by our best scientific theories
    So we should have an ontological commitment to mathematical objects
    If you haven't heard of him, you'll definitely like Max Tegmark. He has a book called A Mathematical Universe and has a few talks on YouTube. He basically postulates what you're saying, that, at the most fundamental level, everything is mathematical, a mathematical structure even. We might live in, and be one big mathematical structure itself.

    A short interview:


    In my personal view, it isn't the abstract mathematical objects that "exist" in some Platonic sense, rather it is the intrinsic/fundamental properties that the mathematics happens to describe that exists.

    Mathematics is invented in the sense that it is a language and we can choose to define it in our own way, so I don't think it "exists". If it did, then we would have to be open to the idea of other languages (and thoughts in general) "existing". With that said, mathematics is also discovered in the sense that we have chosen an axiomatic framework which pertains to our reality (as far as we can tell, with a few tweaks here and there for special cases). We weren't free to choose this, we had to choose it based on what we saw. One could argue that the particular mathematical framework that happens to describe our universe might "exist" - it could be more than just a description/model. If it was, then this would arguably allow for you to go from nothing->laws->universe (physical).

    For example, Newton's law of gravitation and the inverse square law of gravity. It isn't that the numbers or equation exists, or that a quadratic (r², radius squared) exists, rather it is the property that gravity gets much much weaker the further the distance apart that exists. With words (a language), you can't effectively describe it or gain a good intuition of what's going on, but with maths (a better language in this case) you can pinpoint exactly what's going on - perfectly in fact.

    Since we invented/defined the maths to make it useful, it shouldn't be a surprise that it matches what we're describing in physics. With that said, the very fact that the universe can be described perfectly with mathematics (as opposed to can't be) is a mystery. Eugene Wigner: "The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning." - The Unreasonable Effectiveness of Mathematics in the Natural Sciences


    What's amazing to me is that, assuming the mathematical framework is an accurate description of a particular scientific phenomenon, it is often the case that you can simply "do maths" and discover new properties/things about the universe without even doing science. For example, the concept of black holes came out of Einstein's equations of relativity, and in particle physics many particles (such as the Higgs) were a necessary consequence of the maths, and all of these things were only scientifically discovered several decades after the fact.
 
 
 
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