What is the criterion you would give an astrologer? I'm not equating them with astrology, but simply pointing out the countless billions wasted on mathematics that narrows to the rarefied "breed" of pure mathematician earning a salary is as much of a waste as if billions were spent on buggy whip boys. Some things are just meant to fade into museums and history books.
There is no argument that fundamental research (including, but not limited to pure mathematics) is a high-risk investment with a small chance of a large pay-off. But, like most good, high-risk investments the potential pay-offs can be massive and, importantly, not always well-understood in advance.
Would we have GPS without relativity which in turn depends on non-Euclidean geometry, initially considered a curiosity? What about public key encryption, built upon number theory, proudly described as "useless" by the pure mathematician G.W. Hardy? There are so many examples like these.
Be careful of committing the fallacy in assuming that because a clear line of ideas can be traced back from the present day that finding that line was as easy as recounting it.
Out of interest, what is your position of the spending on sport, the arts, and other cultural endeavours?
People are tired of me criticizing maths on here so I'll stop with it.
My opinion on sports, arts etc. is the same. Publicly funded stuff is basically for the elite anyway, it's their way of cleverly siphoning tax-payer money (for there boring outdated interests which they partake in to show the illusion of sophistication). The masses pay to see their interests (and usually heavily taxed for it to boot).
I've had discussion with earnest creationists who point at the fossil record and say "Look - there's no way you can get from A to Z. There's a huge gap." When something is then found that's an intermediate form, they then say: "See! I told you! Now you have two gaps, and it's even worse!!"
So my point is this. I've read what you say, and I can see that you earnestly believe that pure mathematics and the theorems it produces are of no use. I'm asking what it would take to change your mind. What evidence would you need to convince you?
Science is grounded on falsifiable conjectures, and makes progress by testing those conjectures. If you are unwilling or unable to tell us what would be sufficient evidence, then I am unwilling to start chasing ghosts. It seems to me that trying to convince you would be like trying to nail fog to a wall.
Engineering is a critical discipline, and without it, things wouldn't get made. However, engineers rely on techniques that are known to work, and often that knowledge is based on deep theoretical work. Error-correcting codes, via which we get images back from Mars, and which allow reliable communications over cost-effective links are based on theoretical work. Yes, people played about and found the principles, but then they leveraged work done a century earlier to get to the limits. Having done so, they explored the theorems to see what axioms they were based on, and worked to see if those axioms could be circumvented.
It's the interplay between pure theory and pure experimentation that gives us enormous benefits, but it appears that you are willing, even eager, to dismiss fields of which you have no knowledge, purely because you can't see how they can possibly be useful.
It's Blub[0][1], all over again, but here we're not on a linear continuum, we're in a richly interconnected web of dependencies.
Another example. Fourier Transforms were first explored as a purely theoretical construct, showing that functions (with appropriate properties) form a vector space, and that vector space therefore has a basis, and that we should therefore be able to decompose functions into a representation relative to some basis. For decades this was a novelty, and then people started to use it for real. The theorems show the limitations, and then the engineers explore what can be physically achieved within those limits. Wavelets are now often used in contexts where the usual Fourier basis of trigonometric waves prove to be less useful, but the underlying theory is identical, and is still applied. And we know it will work, because of the theorems that were proven decades ago. The vector spaces are, by the way, infinite dimensional, and the work to understand the infinities was done as a part of pure math with no obvious applications.
Another example. We know that some problems are equivalent to others, and we know that the current best algorithms for these problems are exponential. We therefore know, for sure, and not just because of experiments, that some instances of some search spaces will be infeasible without a major break-through. As a specific example of that, multiple times people have claimed efficient algorithms for problems known to be NP complete. In some cases I've been able to prove that their algorithms, while possible useful in general, are definitely not polynomial. I can do that because of the theorems I have to hand.
But you won't care, and I can't make you care. I'm not writing this for you, because you appear not to be willing to change your mind, or consider that a field of which you appear to know very little might just be useful.
No, I'm writing this for people who read your comments and wonder. I'm writing this for people who are willing to entertain the idea that things of which they know little or nothing might be useful in ways they can't yet imagine.
You've given examples of old theorems, and also said they weren't actually useful at first. Later practical work lead to someone seeing use in them (but only the a vague way - limits - which could have been and probably found through experiment as well).
The example you gave of your proof doesn't say all that much, you won an argument (you said those algorithms were practically useful anyway).
I'm sincerely sorry if I come across as hostile. I'd just like to see mathematicians own up and change the corruption from the inside. A lot can come from it, you can change the world (you've taken up a lot it's resources, including some of the smartest people).
You are moving the goalposts again and again, which is why I asked you for the criteria necessary to change your viewpoint. It's clear that you won't, and this will be an endless and fruitless exchange. However, Duty Calls[0].
> You've given examples of old theorems, and also said
> they weren't actually useful at first.
We don't know what of today's work will become useful, or how. I can only give you hindsight evidence, because prediction is hard, especially of the future[1].
> Later practical work lead to someone seeing use in them
> (but only the a vague way - limits - which could have
> been and probably found through experiment as well).
It's limits that are especially hard to establish with experimentation. How do you know that you just haven't yet been clever enough? How do you know that this will always work no matter how far you go? Remember the Ariane 5 explosion? The maiden flight of the Ariane 5 rocket, Flight 501, was lost because engineers reused a unit that had worked flawlessly in every previous flight. The limits were not well understood, and experimentation was expensive. These are guiding principles - sometimes understanding the theory helps more than twiddling bits and seeing what happens. Sometimes twiddling bits is enough. Engineering and Theoretical Research should work together.
In my own work there are at least a dozen cases where knowing a theorem has provoked an exploration, that has then turned out to be useful. The investigation would most likely never have started, the techniques never suspected, without that initial theoretical knowledge. Again, engineering and pure math research in combination, but without the pure math to start with, some things might never be found. We can't know that, of course, but for those who have studied math, the connection is clear. For those who haven't, it's harder to see these things happen. It all seems so obvious in retrospect.
> The example you gave of your proof doesn't say all
> that much, you won an argument (you said those
> algorithms were practically useful anyway).
So, you don't get the idea then. I proved something was impossible given our current state of theoretical knowledge, and that if the claims were correct it was an outstanding breakthrough. He proved nothing, and had a system that worked sometimes, and he was unable to know when, and how, it might fail. I predicted - using pure math research - exactly how his system would fail. I could do something he couldn't, using his system that he created.
> I'm sincerely sorry if I come across as hostile.
Hmm. let's see some of the things you've said:
> ... rarefied "breed" of pure mathematician earning a
> salary is as much of a waste as if billions were spent
> on buggy whip boys.
... and ...
> ... it's the full employment act for pencil pushers.
... and ...
> A lot in common with fundamental religionists.
... and ...
> I never tried getting education in it in the first place.
> I had hunch it's useless and the older I get the more
> I think that was correct.
... and ...
> It's justification for pencil pushers who don't want
> to do real work.
Yes, that does seem hostile. In fact, it seems like you never bothered to study math, and now are trying to convince everyone that something you don't understand cannot possibly be useful.
> I'd just like to see mathematicians own up and change
> the corruption from the inside.
Hmm.
> ... you've taken up a lot it's (sic) resources, including
> some of the smartest people.
So some of the smartest people study math and claim that it is a good thing to be doing. You, on the other hand, claim that it isn't. Your position seems difficult. You claim that something you've not studied, and which is studied and commended by some of the world's smartest people (by your own claim) is, in fact, useless.
Again, I'll never convince you, but I thought your comments should not stand without reply. Perhaps you are in earnest and genuinely believe the things you say, but you are arguing from a position of willful ignorance, which makes it hard to take you seriously.
