When I read the OP I thought to myself I’m going to reply and point you to Roger Penrose’s “Road to Reality” book, the first chapter; but tdvance beat me to it.
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Originally Posted by tdvance
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Roger Penrose suggested that mathematics is "unreasonably effective" because there is a "platonic world of mathematics" that exists independently of human mathematicians (i.e. if aliens did math, they'd come up with the same stuff, though with different names of course!), and that the universe's fundamental laws, whatever they are, are mathematically consistent.
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I would add to tdvance’s post that Penrose also talks about “The Mental World”, this is the picture you have in your mind of how the “Physical World” works and Penrose separates it from the “Mathematical World”. The aha for me was the separation. I build models that fall in the “Mental World” and the struggle for me is building the “Mathematical World”. Don’t get me wrong, I feel the “Mathematical World” is important; it’s what gives the “Mental World” or mental image a chance of surviving.
One of my favorite quotations is Einstein’s:
Physical concepts are free creations of the human mind, and are not, however it may seem, uniquely determined by the external world. In our endeavour to understand reality we are somewhat like a man trying to understand the mechanism of a closed watch. He sees the face and the moving hands, even hears it ticking, but he has no way of opening the case. If he is ingenious he may form some picture of the mechanism which could be responsible for all the things he observes, but he may never be quite sure his picture is the only one which could explain his observations. He will never be able to compare his picture with the real mechanism and he cannot even imagine the possibility of the meaning of such a comparison.
For me thinking of a closed watch (being an engineer) I picture in my mind a set of small gears and a mechanism for turning them. I know the math! I can do all of the calculations to make the hands move precisely. I have the “Mental World” and the “Mathematical World” where they can exactly duplicate the “Physical World” exactly. But what does all of this mean if the closed watch is a digital watch?
Mathematics is a tool.
Jim