Argued ... like a true scientist . We can do mathematics without any appeal to science . However, science cannot function without a mathematical framework. (all 'scientific models' are mathematical models)
> We are born scientists. Children seek to understand the world around them; they do science. It's only later in life that they learn about the power of mathematics to help them reason about the world around them.
Some of us may be born scientists .Certainly not the majority (the same being true for mathematicians , or philosophers ) . How else would you explain religious inclination? I'm sensing you're using a far too narrow definition of the human condition . I for one, as a child , was never to much interested in exploring nature . I was far more focused on internal experience and phenomena . I would stare at a toy for hours (the toy in and of its self may have been boring, I imagine ) , and imagine and construct fantastical scenarios . And I was fascinated and preoccupied with numbers .A vivid memory is that at about the age of five, I arranged the numbers from 1 to 25 in a 5*5 square , according to some random rule . Then , with ave and wonder , as I calculated the sums, I saw that it was what I would later learn to call a 'magic square' . Does that exclude me from your model ?
>"You "assume" that you are part of the world, and you "assume" that you are capable of reasoning about the world."
Not exactly . The ideal observer , for science , has always been 'external observer' the one that would 'see' reality without in any way being part of it or influencing it . Like science, he would have no 'need' for self-reflection . So you don't assume you're part of the world . That's why even so many scientists struggle with quantum mechanics . Anyway , event taking into account these additional assumptions, how does that have any bearing on my argument ? All apples will turn violet tomorrow .
>"Science: first you observe the world, then you build up a conceptual model of the world, then you consider the implications of that model, then you experimentally test those implications."
I tell a scientist my mathematical theorem that no natural number's cube can be written as the sum of two non-zero cubes . How would he go about to test (prove or disprove ) that theorem ? According to 'the scientific method' numbers are just distinct individuals that have no necessary harmony as a whole . So the scientist would , in attempting to disprove the mathematician , build a machine that tests for every number, one at a time, if it can be written as the sum of two cubes . If he finds one , too bad for the mathematician . A likely story. But my money's on the one who has correct proper knowledge of the situation , namely the mathematician . Having an infinite amount of numbers to test , the scientist will never be able to finally prove the mathematician . He would have to be on the negative side, not because of any bearing of reality , but because only a negative answer would finally put an end to the scientist's sisyphean struggle of testing numbers .
Try arguing that induction is impossible with a mathematician .
If the world is, in truth , mathematical, then for a sufficiently powerful and intuitive mind , the whole model-building process would become a futile exercise . Have you ever seen 'The Matrix' ? Compare Neo's vision of The Matrix http://josefflorian.files.wordpress.com/2012/10/large-matrix-blu-ray7.jpg to the scientist, working within the Matrix, building his models and obeying his 'falsifiability principle' . Which one is closer to apprehending the reality of the Matrix ?
>Religion already does that. Religion has made a lot of falsifiable claims , (fixed stars , geocentricism) , and they have indeed been falsified . It also made claims that were essentially meaningless , neither provable nor disprovable . Compare to mathematics/reason, the domain of certainty , who's claims (the infinitude of primes for example ) are as valid now as they were 3000 years ago .
With a few notable exceptions , religion has been the domain of fantasy , not reality . Then again ,religion and mathematics have been, for the most part , disjoint ever since the end of antiquity ("Let no one enter here who is ignorant of mathematics" ,inscription on the platonic academy. )
>It's an argument for its silliness. I find this http://en.wikipedia.org/wiki/Gabriel%27s_Horn silly . Also the existence of a continuous space filling curve : http://en.wikipedia.org/wiki/Hilbert_curve . I also find the experimental violation of Bell's inequality hilarious . Since when has silliness ever been an argument for or against the truth of a statement ? Only a post-modernist would seriously consider it as such. The truth doesn't give a damn about whether or not you find it silly . All modern scientific theories are incomplete and subject to revision .Gödel's theorem is here to stay .
Science is about the journey . A complete theory of everything would leave scientists depressed and without occupation .They would still conduct experiments to try and falsify it, no matter what rationalistic proofs it rests on .It is within their interest to produce as many theories as possible to later falsify. Mathematics is about the destination .