As humans, we created mathematics, this is something we created from "nothing"(?) in order to explain, rationalise things we observe. Now, how can we assume that the universe is logical, and that it can be explained by mathematical equations that WE create?
This is bugging my mind every time I think about it.
Some people are Platonists and think that mathematics exists as a sort of idealized mental realm (similar to Plato's theory of forms, but more reasonable in this context). If this is the case, we would say that mathematics is mind-independent and trans-universal; mathematical truths are true regardless of whether we exist to observe them, and they are true in all possible worlds.
Others think mathematics is a linguistic "game" played by man, that it is mind-dependent and completely of our creation. Still others think mathematics is a set of derivations within a "formal system" (or formal language); this is sort of a middle ground, as it implies that mathematics is a construct of man but that it has a weaker mind-independence/trans-universal property, namely that any entity in any universe examining the same formal system comes to the same conclusions.
But as I said, how you feel about this question has no effect on the real-world practice of mathematics. Most mathematicians you ask will have some opinion on the matter, but it's not like you can say "I'm a Formalist so I don't believe in that theorem." The human process of doing mathematics and the validity of theorems with respect to their assumptions has no relation to the questions of the ontological or metaphysical status of mathematics as a whole.
All we can do is observe that the behavior of an object in the physical world behaves as the model predicted. With enough observations and enough predictions, we can say with some certainty that the model is accurate. We invented the model, then we tested it and confirmed that it approximates reality, usually within a known range of uncertainty.
The universe is still logical in that causality is relatively logical. We're just describing the universe as we see it.
You might also want to read about Euclidean and Non-Euclidean geometries and the history of the parallel postulate for how humans have grappled with this question in the past, and how our thinking has evolved in the last 150ish years. This is a fantastic book with a mix of math, philosophy, and history:
http://www.amazon.com/Euclidean-Non-Euclidean-Geometries-Dev...
(I'm sure you can find a used copy for a lot cheaper than that link!)
It's sort of an open question whether mathematics is invented or discovered.
"The opposition between the analytic and the synthetic approach to mathematics in the first half of the nineteenth century is well-known.. Less well-known is that both distinctions were rooted in a cultural clash, in the period of the first French Republic, between the established analytical tradition, guided by Lagrange and Laplace, and a new, geometrically-oriented approach, swayed by the revolutionary upstart Monge. These mathematicians had been assigned by the government to normalize and rectify mathematics to a perfectly transparent and hence universally learnable 'language'"
Gaspard Monge was the Director of Ecole Polytechnique, the pioneering French military engineering school, educational predecessor of West Point, MIT and others (http://www.uh.edu/engines/asmedall.htm).
"Monge... expressly rejected the reduction of mathematical reasoning to a formalism. He insisted on the indivisibility of form and content, and denied that any rules, mechanical or otherwise, could be given for the conduct of mathematical investigations. For him, analysis was not a language, closed in itself, but merely the 'script' for the notation of reasonings about quasi-empirical, especially geometric contents."
As humans, we created mathematics
I'm neither mathematician nor philosopher, but there seem to be some a priori truths in mathematics (I hope I'm using that term correctly) which are not defined or created by us: the ratio between a circle's circumference and diameter (Pi), the distribution of prime numbers, etc.
If you are bugged by where those come from, then I don't think you're alone!
> unless your argument is that mankind's idea of God also corresponds to the reality of what exists outside our heads
But all men do not agree on what God is at first place, or whether he exists or not. It doesn't make the idea of god less powerful(that's my point), however if you define "reality" as what men can experience, no men has ever experienced "God" and created a reproducible experiment of God. Therefore "God" cannot be a reality today. God is a philosophical question.
No, in that God is not (just) a philosophical question. There is a reality which exists. Either there is a God who actually exists in reality, or there is not. It is a question of what is the truth about what actually exists, not just a philosophical question.
Well, which/what "God"? the definition of God itself is purely a philosophical question. You can't ask yourself whether "God" exists or not, until you define precisely what you mean by "God". And men do not agree at all on what "God" is, even those that claim they follow the same religion. That's why you can't ask "science" to answer a question on a matter that has no formal definition to begin with. God is an idea first and foremost. a vague idea at best that bear no exact definition, thus a philosophy.
Mathematics is not the real world; it's a map, not the territory. But of course it's not surprising if a map resembles the real world...
https://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.htm...
Therefore the practice of mathematics can only take place in a universe where universal, repeatable, rule-following procedures are possible. It has effective regularity in the cosmos as a dependency.
Looked at this way, the thing to be surprised at is not so much the effectiveness of mathematics at describing the universe, but the universe's admission of regularity - once that is granted, both mathematics and its effectiveness seem to follow unproblematically, imo.
Perhaps phrased in a more rigorous way, the universe is a spectacularly huge and complex system, and a single human brain is a positively tiny piece of that system, which is very much a part of the system and not external to it. So the question really becomes something like this: to what extent can a tiny little piece of a huge system contain within it a fully functional model of the entire system, which inevitably means it contains a model of itself?
I'd argue a number is just an idea, and all ideas exist whether you've thought of them or not.
Mathematics is not unreasonably effective any more than English is.
Some of these patterns appear so fundamental that it's virtually impossible to imagine them (in a detailed way) as being different. Maybe this is because our brains, being physically embodied, are constrained by these same properties of nature in terms of how they process information. We can't picture 1+1=5 because we literally can't think it.
Once you have a language with a grammar, you can also start exploring the "pure" properties of the language. Writers do this with written languages, constructing oddities like Ulysses and "a rose is a rose is a rose" and buffalo^8: https://en.wikipedia.org/wiki/Buffalo_buffalo_Buffalo_buffal...
Since the language was induced from nature, exploring its structure can sometimes let you deduce very strong and powerful hypotheses about nature. But these hypotheses are not true (a.k.a. theories) until they are confirmed by experiment or observation. The combinatorial rule space of a language is so large it will contain an effectively infinite number of meaningless coincidental patterns, and Turing proved that the halting problem is undecidable so you can never be "done." The inverse is also true: there will always be facts of nature that cannot be hypothesized by studying any language or logic system. This is the primary consequence of Godel's incompleteness theorem.
TL;DR: Languages are induced to describe reality, therefore you can make conjectures about reality by playing with them. But not all such conjectures are true, and some things must be true that cannot be thus conjectured.
Edit:
Come to think of it, I am not certain that "The combinatorial rule space of a language is so large it will contain an effectively infinite number of meaningless coincidental patterns" is true, or at least I'm not aware of a proof of this akin to Godel's theorem (wouldn't this be the inverse of Godel's theorem?). It could be that all theorems and patterns in mathematics (and other languages for that matter) either directly reflect something in nature or are isomorphic with something that reflects something in nature.
If this were true it would be impossible to make a meaningless statement that is syntactically correct in any language. Tolkien's endless discussions about orcs and elves are talking about something, just maybe not literal orcs and elves.