> The view that mathematics should try to constrain itself to physical > reality is, in my opinion, not a crackpot position.
It does seem to me that - as stated - this is precisely a crackpot position. The problem is none of the words "physical", "reality", or "constrain", the problem is the word "should". In practice mathematics, as referred to by the overwhelming population of mathematicians, means the abstract study of formal patterns, and that's all. The mystery of it all is the way in which the patterns investigated by abstract mathematicians turn out to apply so directly to aspects of the real world. But it's going to be hard to develop these abstract theories if you have to be consulting one of these "real world characters" [do I have to call them other than "cranks"?] to see if _they_ happen to think what you're doing is OK.
What about geometry? Do we really have an absolutely clear answer to what the geometry of the physical world is? If I can assume not, then what is anyone to do? Can't study Euclidean geometry, because it might turn out not to be maths; similarly can't study non-Euclidean geometry either. In fact, the maths department is at the mercy of the physic department to inform it (eventually, perhaps) what it may or may not include in the curriculum. Clearly this is insane.
I am no expert here, but I looked at the OP's rant, and it appeared to me to be classic crankery. How else can you interpret it but that he thinks "mathematics" means "calculating things (in the real world)"? I see he has also written a book - can anyone make any sense of it? Again, there are all the standard crank signs: cranks do not investigate some new idea, they "replace" the bit of maths that's beyond them with something usually half-baked - and even if it could be formalised into something coherent, it's very far from clear that it achieves anything other than applying a patch to the crank's pet peeve. I mean, OP "replaces" trigonometrical functions with something "better": can that ever be coherent? If the "replacement" is different, it has not replaced anything, since the original functions still exist; if it's just a different description of the same thing, this hardly counts as "replacement" either.
I'm also puzzled by a quite simple question: if you reject the axiom of infinity, then you have no infinite sets. Can you still say - for example - that the integers form a group under addition? (How? By some grotesque circumlocution? I don't know...) Do you not simply lose all algebra relating to infinite sets? When people like Tony Orlow babble on about "numbers", it's easy to understand why it doesn't bother them that their creations have more or less no well-defined algebraic properties at all (and I assume that they have simply no grasp of what they are missing, in their headlong flight to create their "anti-Cantor" patch), but how can someone apparently involved in teaching maths resolve this? (This is partly a genuine question: I know it's perfectly legitimate to investigate set theory with an axiom of anti-infinity, but I just can't quite imagine how you proceed. Presumably you can define the naturals using more or less Peano's axioms; you can show therefore that there is no maximum natural; you can see that it cannot be possible to count the naturals against a ditty that stops anywhere, yet somehow you can't actually talk formally about "all the naturals". ??)
> Let me use an example. The mathematicians Wiener and Banach lived at > about the same time. Banach chose a much more abstract direction to > take his mathematics, looking at general properties of infinite > dimensional normed vector spaces beyond Hilbert spaces and spaces of > continuous or measurable function. Wiener looked more to real life > applications, and building on the work of Einstein put the phenominum of > Brownian motion onto a rigourous mathematical foundation. Many very > talented mathematicians have studied Banach spaces, but it has proven to > be a remarkably barren field. On the other hand Wiener's approach has > led to the rich theory of stochastic calculus, which has proved > extremely useful in studing the heat equation, harmonic functions, and > even economics.
That's an example (or rather two examples) about which I know next to nothing. But another example is the use of finite fields in encryption. Surely (in Hardy's time, say) if a Petry or any other of these "Should" freaks had come along, finite fields would have been a prime candidate for banning, since they "had no relation to the physical world". The history of maths is all about people wandering off and investigating something just because they wanted to, constrained only by the need for mathematics to be, um, mathematical. That is, to proceed by axiomatic, logical argument (exactly the sort of thing that the cranks never seem to manage).
Actually, I'm not sure that your examples above are fair anyway. Consider the relative contributions of fortune-telling and astronomy to the everyday happiness of the proverbial rider of the Clapham omnibus. Fortune-telling provides lots of entertainment, keeps proles out of gin palaces, provides employment for considerable numbers of people (at least, around here it does), and generally forms a part of the social fabric. What has astronomy done? Sell a few telescopes to schoolboys? Of course I have no doubt which has the greater fundamental value, but I do not think this can be assessed by the sort of naive economic argument you seem to be relying on.