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You can simply dream up whatever axioms, undefined terms, and rules of logic you want. However, one runs the risk of having an inconsistent system or a system that is not interesting to others. Godel's Incompleteness Theorem does not say that this can't be done. Furthermore the "underlying contraints" imposed by Godel's Incompleteness Theorem is not at all what most mathematicians study. Unless I'm misinterpreting your meaning here.

There are knowledgeable people who do not believe that mathematics is independent of our minds. It's not too far fetched of an idea. While I do not personally agree with this, I won't downplay such beliefs.

>You cannot simply dream up whatever structure you want and have it mean what you want and behave how you want. See Godel's incompleteness theorems.

That is not at all what the Incompleteness Theorems actually say. They say literally nothing whatsoever about what sorts of structures you can implement inside a given foundational theory, except that there will always be more, because given any foundational theory, you can construct two more foundational theories as extensions (one in which the Goedel statement is unprovable, and one in which the theory believes it's inconsistent).

>Mathematical truths and objects are real things with existence independent of our minds

What makes you say this? Isn't this an open philosophical question? What makes you say that mathematical objects exist independent of our minds? I can dream up a set of axioms of my own and do maths from there, so I don't think mathematics necessarily exists in some Platonic ideal dimension independent of our minds.

>Mathematical truths and objects are real things with existence independent of our minds that we "discover," not just designed things.

While its almost certainly true that the content of mathematics is mind independent, it is far from obvious that these objects are "real things".The real meat of the issue is how exactly the mind-independence is cashed out. Different ideas paint a vastly different picture of mathematics and even the universe. For example platonism vs. nominalism. Lets not be so quick to put forward as an obvious truth the critical issue in question.

Can't a mathematical theory compress, generalize, and map out many relevant empirical facts very well without needing ontological commitments to the generalizations themselves?

The real numbers seem to be a perfect example: if you work in physics at scales where quantization doesn't noticeably apply, the only way to calculate correct predictions is really to use real numbers and continuous (mostly Euclidean) spaces. But that doesn't mean physical objects are ontological shadows of our mathematical abstractions, as Plato's Allegory of the Cave portrayed it. Quite the reverse: when you get down to a sufficiently small, fundamental level, objects, space, and time stop being continuous and correct experimental predictions only come from using discrete formalisms.

You can then proceed to ask, which one is Platonically real, the continuous mathematical spaces or the discrete physical ones? But I think the answer there might be, "Who says anything is Platonically real? The map is not the territory, so shut up and calculate."

>Can't a mathematical theory compress, generalize, and map out many relevant empirical facts very well without needing ontological commitments to the generalizations themselves?

Maybe, but its not obvious. The fact that the same generalizations are multiply realizable in different processes/structures certainly says something interesting. The consequences of this multiple realizability hasn't been fully investigated.

Your response seems to be arguing against my post which was mainly about mind-independence, by arguing against platonism. I don't see that mind-independence necessarily implies platonism. In fact, I find all forms of platonism extremely distasteful.

>The map is not the territory, so shut up and calculate.

Right, but this in fact goes to the heart of the question of the philosophy of mathematics. When someone says that mathematical objects are mind-independent, they are not talking about the notation itself (the map), but rather the content of the notations (the territory), i.e. the structure revealed through the notation. It should be pretty obvious that there are many interesting questions about the mind-independent structure of the territory. "Shut up and calculate" isn't an answer to this question, but rather the attitude that the answer simply doesn't matter. For many fields the answer doesn't matter, but the question is worth asking nonetheless.

I don't think math exists without sentience, it is a construct . What math describes can and is "real" in the traditional sense, but that doesn't necessarily make objects, concepts in mathematics "real objects." I take this position with language as well, it's all a metaphor.

But observations of real world systems can be identical to a specific mathematical system. I.e. the time it takes for a thing to fall at specific gravity at specific height, the frequency of a particular pendulum, and so on...(i.e. the rest of modern physics).

True, it is our observation and our model which are similar, so I suppose the philosophical question then is up to what point we can trust our observations. And if we trust our observations, I would conclude that the similarity of our observations and math means that the real world can at least exhibit 'maths', which means our minds are not the only place where math can exist.

The smartphone I'm typing this on leads me to conclude that lots of our observations are highly trustworthy :)

Yes, I think that humans have some ways to make sense of the external world, and math is one of them. May not be highly developed, could be buggy. Probably lots of things we could never figure out, like rats who can't solve mazes where they need to turn at prime numbers.

Our math sense could even conflict with our other useful facilities (as is the case with how easily fooled humans are when it comes to statistics).

>He is probably a formalist.

It's actually a she =)

Anyway, I do agree with you (and the author) that mathematics has the potential of being a superb pedagogical vehicle in teaching design thinking.