You claim that Peano Axioms are ‘definitional’ and thus not true

That seems like a rather exotic reading of what I wrote.

In addition, your idea of the independence of axiomatic systems fails as a challenge against Platonism for two reasons.

That’s an odd comment. I have not attempted to challenge platonism. I have merely explained why I am not a platonist. I do not criticize others for being platonists.

Another thing I would like to point out is the inability for fictionalists (which I think is a view that borders on the absurd) and nominalists to ascertain why mathematics is so useful to science without simply stating it as miraculous.

May I suggest that you tone down the rhetoric.

I have explained, in other posts, why mathematics is useful in science. Try “Why mathematics is useful to science” as one such post. You might not agree, but I have at least provided an explanation which did not suggest anything miraculous.

Regardless, your arguments against Platonism fail to deliver in a pretty severe way.

I’m sure that is true. As we mathematicians might say, it is vacuously true. That is to say, I have not made any arguments against platonism.

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]]>In addition, your idea of the independence of axiomatic systems fails as a challenge against Platonism for two reasons. 1) Many mathematicians still believe that there could be a single mathematical universe that would solve the ‘independence’ problem associated with several mathematical problems, including the continuum hypothesis (Hugh Woodin has done interesting work on this with Ultimate L). 2) In addition, one could think that there are many different systems of mathematics and still be a Platonist with regard to ontology (Joel David Hamkins is one such person). These individuals might tend to think that different axiomatic systems are as real as the different non-euclidean geometries that currently exist.

Another thing I would like to point out is the inability for fictionalists (which I think is a view that borders on the absurd) and nominalists to ascertain why mathematics is so useful to science without simply stating it as miraculous. Yea, I often hear the retort of “Math is simply representation and we create it.” That type of argument fails for two reasons. 1) If we think of math as a representation, then the thing it is describing should have an existence that is equivalent to some sort of mathematical structure (the one represented). Did we invent the patterns in nature? Do the predictions that pure math has made in the sciences (especially theoretical/particle physics) that were later empirically verified really make one feel comfortable with the assertion “Don’t worry, its still all a constructed narrative.” Why doesn’t anything else labelled as fictitious (novels, movies, etc.) describe the world so accurately, what gives math a special distinction?

You even seem to point to some form of structural realism (easily compatible with mathematical realism) when you make this statement: “And if they can find a sufficiently structure mathematical representation space, then they can use that mathematical structure to analyze representations, and to make plausible inferences about the world that is represented.” Mathematical Platonism doesn’t really refer to a heaven with perfect, timeless, abstract entities. There are several different branches of mathematical realism that are determined why the ontological nature of mathematical entities themselves. Many structural realists actually forgo mentioning particular objects and instead believe that the mathematical structures themselves are all that exist.

Regardless, your arguments against Platonism fail to deliver in a pretty severe way. In fact, I think fictionalism is extremely hard to justify/take seriously relative to the history, practice, and results that mathematics has garnered humanity over the centuries.

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