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October 15

Rational trigonometry

The mathematician Norman J. Wildberger writes this lemma here (at 04:48), but does not prove it. Can anyone help? יהודה שמחה ולדמן (talk) 17:56, 15 October 2017 (UTC)[reply]

Let the sides of Q be x and y. Then the area of Q is xy, and the area of Q₁ and Q₂ are x² and y². Multiply everything up (remember (xy)² = x² y²) and you get the formula in the lemma. Staecker (talk) 18:14, 15 October 2017 (UTC)[reply]

Now it occurs to me that Wildberger is trying to argue everything without ever using distances. In that case I'm not sure how to prove the lemma. Staecker (talk) 19:35, 15 October 2017 (UTC)[reply]

My advice is to ignore him. He has some rather cranky views, and it's probably not worth your time. --Deacon Vorbis (talk) 21:37, 15 October 2017 (UTC)[reply]

Wildberger is one of the few mathematicians who appreciates that mathematics has been contaminated by pseudoscientific notions about infinity. The problem is not that these notions can't be treated rigorously within mathematics, they can be so it's not that the math is wrong, rather that it becomes unnecessarily convoluted. This doesn't mean that Wildberger's approach is optimal, but it's worthwhile to think about these issues instead of treating everyone who is an Atheist w.r.t. the God of Infinity as an outcast. Count Iblis (talk) 22:43, 15 October 2017 (UTC)[reply]

Math has no scientific notions, psuedo- or otherwise. And to quite the contrary, Wildberger argues precisely that infinite sets cannot be treated rigorously; he claims that logical contradictions result from the axioms of set theory. However, when pressed for details, at best he'll evade the question, or at worst (especially when talking to someone who doesn't have any formal training in math), he'll try to dazzle with dishonest ~~bullshit~~ nonsense. This is what makes him a crank. If he wants to reject infinite sets on aesthetic grounds and develop stuff without them, then he's more than welcome to, even if the vast majority quietly ignore him. But his continual claims that everyone else is wrong—that's why he's a crank. And also, your religious metaphors here are completely inappropriate. --Deacon Vorbis (talk) 23:05, 15 October 2017 (UTC)[reply]

From an abstract point of view, math is totally isolated from science. However, there is a reason why mathematicians have chosen to work within some framework. Why were real numbers defined in the way they actually were? There are intuitive notions behind this that came for classical physics as practiced in the 19th century. If math were totally disconnected from science, it would have been a lot more cumbersome to use math in science. Now, the modern view in physics is that the continuum should be understood as a limit, it doesn't exist a priori in a physical structure, see e.g. the third paragraph on page 12 here. Had this been known back in the 19th century then it's likely that topics like analysis would have been set up in a different way involving the modern notions of scaling and renormalization as we do in physics.

Now, Wildeberger is talking to a lay audience on YouTube, he has to sell his ideas there and that's not going to be done successfully if he were to stick to the orthodox way of doing things, no one would watch his videos. In this day and age, you need to be provocative or else you'll not be heard (which is why Donald Trump is president instead of a more reasonable person). Count Iblis (talk) 00:43, 16 October 2017 (UTC)[reply]

For some reason, I seem to recall that 't Hooft is a bit of a fringe guy himself, although I don't really know this stuff. (I did notice in that paragraph that he's using "dense" incorrectly though, at least from a mathematical terminology standpoint). But I don't really know what your point is. Wildberger isn't even happy with rational numbers, or the naturals, or any infinite sets, let alone the reals. And like I said above, his objections run a lot deeper than just aesthetic or utilitarian ones. And being provocative doesn't excuse being dishonest, which I've encountered from him personally while trying to engage him in the comments on those very videos. Anyway, this is getting farther and farther off-topic, and I suggest we just stop here. --Deacon Vorbis (talk) 01:40, 16 October 2017 (UTC)[reply]

I've heard invective against the reals, the rationals, the negative integers, and zero, but never against the natural numbers, and I admit I'm tempted to watch one of his videos just to see what objection he could possibly have. OldTimeNESter (talk) 03:06, 16 October 2017 (UTC)[reply]

See here. Count Iblis (talk) 03:15, 16 October 2017 (UTC)[reply]

I have already noticed a while ago that the guy is just weird.

And even so, what if the area of a quadrance is irrational? How would he solve that? יהודה שמחה ולדמן (talk) 04:28, 16 October 2017 (UTC)[reply]

See rational trigonometry. Count Iblis (talk) 08:37, 16 October 2017 (UTC)[reply]

I've seen some of his foundations videos and I'd say that while his pov is not mainstream, it is interesting. Not only does he refuse to accept the concept of infinity, but he doesn't even accept very large numbers. It does seem reasonable to ask whether the smallest prime greater than π⋅10^10¹⁰⁰ is actually a number; it could never be written down as a decimal because the universe too small to contain it, even if it were possible to actually compute within the lifetime of the Milky Way. But while he does raise some interesting points, he's often dismissive of other points of view, which, as Deacon Vorbis was saying above, is the kind of thing that takes his ideas out of the realm of serious philosophical discussion. My main objection (in case anyone cares) is that at some point you have to just declare that you've done due diligence on the philosophical issues, just get on with with the math, and start delivering results that someone outside the field might find interesting and useful. You can argue about whether real numbers exist but I doubt that many physicists think the fine-structure constant isn't one. --RDBury (talk) 09:44, 16 October 2017 (UTC)[reply]

I wonder whether he thinks the fine structure constant exists as a number in mathematics. Maybe it can't be calculated using an algorithm. Do we then have physical numbers that can be measured and mathematical numbers that have been computed. Then why would we be entitled to multiply by it etc. etc., lots of stuff to get us invited on a tour. Dmcq (talk) 12:53, 16 October 2017 (UTC)[reply]

OK, that is enough... I totally lost you guys. So does he fit the bill of a crank or not? יהודה שמחה ולדמן (talk) 20:19, 16 October 2017 (UTC)[reply]

Crank. E Eng 22:47, 16 October 2017 (UTC)[reply]

The precise way things work in physics requires one to regularize things anyway, you have to impose a cut-off for fluctuations below some length scale which makes the theory de-facto discrete, you can in fact define the model to be a lattice model (but that's not a mathematical convenient way to go about things). Physical observables are obtained as the scaling limit of such a theory, the cut off is eliminated by an elaborate limit process. Classical physics has to be understood as something that arises in such a limit. But back in the 19th century the notion of a continuum that ultimately comes from the physical world came without this prescription that it is actually a result of a limit procedure, because quantum field theory was not invented yet. Mathematicians then found a way to deal with the continuum directly, but this is perhaps not the best way to go about things.

Since we live in a digital world today, we deal with the fact that smooth looking pictures are in fact made out of pixels. Children who play with their smartphones all the time, know very well that if you magnify a picture a lot you're going to see a block-like picture. There is then no reason to introduce the problematic concept of a real continuum to teach calculus. We can just as well introduce calculus based on a notion of coarse graining and scaling limits applied to a discrete system, where you get smooth functions in some appropriate scaling limit. Count Iblis (talk) 20:56, 16 October 2017 (UTC)[reply]

A child looking at a newspaper wirephoto in 1935 knew the same thing. The poor OP! E Eng 22:47, 16 October 2017 (UTC)[reply]

First, that's nonsense. Students generally don't see a rigorous characterization or construction of the reals before they learn Calculus anyway. But without them, you don't have things like the Intermediate value theorem, or the Mean value theorem, or any number of other fundamental results. You can still get something similar, but trying to explain it is going to be more complicated, and there's no good justification for all that extra complication, especially when it's not important for the basics. There's also nothing "problematic" about the reals; they work just fine. And second, this again has nothing to do with Wildberger's cranky ultrafinitist insistence that everyone else is wrong. --Deacon Vorbis (talk) 23:27, 16 October 2017 (UTC)[reply]

Every year or so I drop in to see if the Ref Desk is still as wacky as I remember. It is. I'm unwatching; see you in a year or so. DV, why do you waste your time here? E Eng 23:37, 16 October 2017 (UTC)[reply]

Reals work fine, but then anything we can every do in math is only ever going to involve a finite number of manipulations using a finite number of symbols. So, we can give an interpretation to what the symbols are supposed to mean, but the real beef is in the rules for manipulating the symbols, so ultimately it is all just discrete math anyway. Count Iblis (talk) 01:36, 17 October 2017 (UTC)[reply]

Dammit, I forgot to unwatch. I won't make that mistake again. E Eng 01:38, 17 October 2017 (UTC)[reply]

Iblis, there are some arguments to be made for ultrafinitism, but not that one. That one is complete and utter nonsense. The manipulation of symbols is one thing; what the symbols refer to is another thing completely. ---Trovatore (talk) 01:53, 17 October 2017 (UTC)[reply]

But that leads to another problem. Suppose we do something that falls unambiguously within the domain of discrete math, we simulate a cellular automaton (CA) and we see some patterns appear and we can then attempt to explain these patterns that appears to be have universal properties . Then surely it would not be reasonable to invoke uncountable quantities to explain what we see? But in theory, a large enough CA could well have ended up simulating a virtual mathematician doing real analysis. Count Iblis (talk) 19:19, 17 October 2017 (UTC)[reply]

First of all, it absolutely can be reasonable to invoke uncountable quantities to explain, or at least analyze, the behavior of discrete and finite objects. That's exactly what analytic number theory does all the time.

In your last sentence, you're making the same mistake you did before — conflating the representation used by the observer with the thing observed. --Trovatore (talk) 21:59, 17 October 2017 (UTC)[reply]

Going along with the pixel analogy - why do we bother with SVG then? They are, after all, only infinite "approximations" of their renderings on finite pixelated screens... 93.139.2.24 (talk) 01:58, 17 October 2017 (UTC)[reply]