I responded to Eugene Wigner's famous paper by taking a position that human mind matches patterns in Physics and Mathematics (treating the latter as nothing more than a formal game) - http://tech-maths.blogspot.co.uk/2017/02/effectiveness-of-mathematics.html

However, are there certain Mathematical spaces that are real? For example, consider solutions to Einstein's Field Equations. If massive objects are contained in and interact Minkowski's space, then is it empirically real? Please do not give a general philosophical opinion but an answer that is logical and ideally falsifiable.

  • If (physical) objects are real and they "are contained in and interact Minkowski's space", then mathematical spaces are real. – Mauro ALLEGRANZA Feb 7 '18 at 7:39
  • What you are suggesting, the "mental reality of concepts represented by symbols", "indeterminate physical reality" underneath, which "produces logically related concepts in the mind", sounds like Kant's phenomena, noumena and categories of understanding. After Kant this idea ran into big troubles: why and how "mental reality" is sharable and not subject to whims, how "indeterminate" reality manages first to determine theories and then force their revision... On your/Kant's account physical reality is "indeterminate", so obviously there is nothing we can know about it and no math can be real. – Conifold Feb 8 '18 at 0:50
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    Can you explain what you mean by a mathematical space? For example a group is an abstract mathematical object. But the collection of ways you can rearrange a set of five objects certainly exists. It has a deterministic answer and you can demonstrate these permutations to anyone's satisfaction. Are you asking if abstract objects are real? – user4894 Feb 8 '18 at 2:44
  • I regard reality itself to be a mathematical space, although we don't know all of its parameters. If you believe in Planck length/time, any mathematically continuous space is false, but it's still often convenient to pretend spacetime is continuous. – user935 Feb 8 '18 at 16:56
  • @barrycarter First of all: 'If you believe in?" Physics by fairy-dust? Second: There is no conflict between a space with indistinguishable points, and continuity -- consider the 'monads' of Non-Standard Topology. Planck-length just forces upon us an 'Ultrafilter' that is not transfinite, but contains all reals below a given size. – user9166 Feb 8 '18 at 19:37

Physical theories are mathematical models that approximate reality. If the models of any science were real and not representations, or even if they were accurate representations, that science as a whole would simply be finished with its job. The fields of the field equations aren't real and neither are the mathematical tensor spaces the operate on. Space is real, and they are abstract models of how it behaves.

So being used by physics does not make math real. On the other hand, from an intuitionistic point of view (a la Kleene), mathematical models are as real as your vision or your emotions are.

Mathematics and logic are patterns of communication to which humans naturally respond. Euclidean space has the same level and kind of reality that fear does. Is fear real? Well, that depends... But however you handle fear ontologically, math goes in the same bucket. And intuitionism affords that bucket the status of reality. It is real as a set of patterns, and those patterns can be empirically validated or falsified by exposing them to humans.

Four-dimensional space has the same level of reality as other patterns of reaction to other stimuli, say, the fear of spiders. If you are one of the people who share and understand it, it produces a given feeling. In the arachnophobe's case, anxiety of a common form, in the mathematician's, a sense of settledness that is common to most mathematical propositions.

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