Tropers that do NOT hate math

I like math. A lot. Not always the greatest at it, but it boggles the mind when I hear people talk about how they'll never need it in real life.

Well, yeah, if you're not taking a math-based job you won't need calculus, but you probably won't need history or gym either. THIS IS A CLASS TO EXPAND THE POWERS OF YOUR BRAIN. USE IT.

Math is amazing. It makes me sad that so many of the teachers are so bad, because most of the people I tutored who said they were bad at math just never had it adequately explained in a way they could easily grasp and were fine when they did.

I don't hate math. Then again, I am a son of a math professor.

I only know Year 10 stuff, and even then not very much of it, but I'm looking forward to learning more. Might help with macroeconomics...

I like math, and I'm good at it. I like the logic part; even if I haven't been specifically taught how to do something, I can usually use logic to figure out how to solve the problem.

Though I'm no where near the level of you guys. Probably because I'm only in 8th grade.

edited 8th Jun '11 7:27:30 AM by Crewe

I used to hate maths. Probably at least partly because my maths-teacher dad was pretty pushy about it. I was Ok at it, but used to claim that my brain didn't "work that way".

Oh dear...

So: I didn't study maths at A-level, although I picked up some stuff in Physics. Went to Medical School. Realised that statistics is actually freaking awesome once you have something interesting to apply it to. Then realised that Computational Neuroscience is awesome. And then realised that you need an excellent maths background to become a computational neuroscientist. :(

So now I'm playing catch-up for something I had a lot of opportunities to learn before. Hopefully the MIT website will help a bit - I really want to start understanding this stuff a hell of a lot better.

You're studying to be a computational neuroscientist? Very cool.

It's pretty crazy that a bunch of simple sigmoid-of-weighted-sum functions end up Turing-complete.

edited 9th Jun '11 2:41:14 PM by Tzetze

Yes and a thousand times yes! I did some computational neuro courses this year and (grades permitting) I'll be starting my Ph D in October. I'm insanely excited. But it means I need to give myself some major maths study time over the summer.

You may already know this, but it blew my mind when I found out: The Game Of Life, when given a large enough surface area, can be Turing complete.

An infinite surface area, you mean? Since Turing machines have infinitely long tapes.

I think the coolest thing I've seen in the Game of Life is "faster than light" travel. Demonstration in this lecture somewhere.

You just blew my mind. And also started a fight between me and my boyfriend over whether it makes sense to describe translation of life objects as "movement" with "speed" and if describing the transporter as an FTL device is disingenuous or not.

I don't think Turing machines have infinite tape - infinite tape would be a Turing oracle. The minimum requirements for Turing completeness are actually quite simple. My understanding is that it's difficult to build this in Life, but I'm pretty sure it's been done on a finite surface.

Well, from a purely materialistic perspective I'd say that the human mind is a finite state machine, and that therefore it is not Turing complete.

There is a finite, if huge, number of configurations that a human brain can assume; and this, for example, means that it cannot count up to arbitrarily high numbers. In a certain sense, this means that the Pumping lemma applies to human minds: we are not capable of recognizing the set of all strings of the form

a a a .... a b b b .... b

where the numbers of "a" and "b" are the same, because we are bound to "lose count" after a while — for example, after the number of "a"s exceeds the number of brain states that we can use to represent numbers.

The finite-surface Turing machines implementable in the Game of Life, as well as the computer that you are using, are only finite state machine approximations of a "true" Turing machine. They are not such a machine themselves: for example, there exist computable algorithms that your computer could not execute, because it would "run out of memory", and the same would happen with your Life machine.

edited 11th Jun '11 12:29:00 AM by Carciofus

Yeah, abstract Turing machines have infinite tape in that you can always move another square to the left or right if you need more space. Wouldn't that be nice, not having to bother with that silly garbage collection business

As for neural networks I was thinking of this and a(nother) mention on Wikipedia: "Work by Hava Siegelmann and Eduardo D. Sontag has provided a proof that a specific recurrent architecture with rational valued weights (as opposed to full precision real number-valued weights) has the full power of a Universal Turing Machine[4] using a finite number of neurons and standard linear connections.", citing this, which unfortunately I haven't read.

Well, that's the sort of thing we need to make a generalized abstract mathematical definition of then, eh?

edited 11th Jun '11 12:45:45 AM by Tzetze

On the topic of cellular automata and the like: does anyone know any good textbooks on the topic? I once had a look at Wolfram's "A New Kind of Science", but I admit I was a bit put off by the title and by what I saw as a lack of rigor and hard results — it's been a while, but from what I remember it seemed to me that the bulk of that book consisted in simulations of many different cellular automata and in informal discussions of their properties, plus a bunch of somewhat unsubstantiated claims about how cellular automata were going to revolutionize... well, everything.

EDIT:

A doubt, however, that comes to my mind is that that sort of system might perhaps end up being very unstable with respect to random fluctuations: after all, the rationals are dense in R, and this means that any error — no matter how small — might end up changing completely the state of the system. This would make implementation rather hellish, I think, although perhaps not impossible in principle — I dunno, perhaps a combination of extremely low temperatures and error correcting codes could work?

edited 11th Jun '11 1:08:04 AM by Carciofus

On Turing completeness: OK, really out of my depth, but I'm really sure that being "Turing Complete" is not the same as being a "Universal Turing Machine". A Turing Complete machine simply has to be capable, in theory if not in practice (needing infinite time and whatnot) of performing all the relevant small calculations that can add up to larger computations. At least in theory, the simplest Turing machine would be capable of simulating the most complex one, giving enough time and clever code. In a MIT computing course I took online, the Turing complete "set" was described as 6 abilities. Addition was one of them, but I can't remember what the others were.

The paper looks very interesting - it has gone into my to read pile.

So, without having read the paper at all - what if you implemented it with neurons instead of with electronics? Either simulated neurons (which I believe would work too), or real ones? Neurons are huge (compared to what computer scientists work with, anyways), each "bit" of signal is dependent on massive ion channel involvement. Noise is still a problem, but that's probably the reason that neuronal size doesn't decrease as quickly as brain size increases throughout our evolutionary history.

Well, floating point on computers basically sucks, but a neuron can either fire or not fire, it's not like having a slightly higher value in one of its inputs would mean that it fires more, or the reverse.

Well, that's the important thing, an actual machine can run out of memory where the abstract one can't.

I don't usually see machines referred to as Turing complete, anyway, but programming languages, which are generally understood to be implementable on all sorts of hardware anyway, and which have properties independent of what they're running on.

Yup, my bad: Turing complete is more relevant to programming languages.

I suppose you'd know better than me, but I meant the intensity of a given firing, not the amount of firings.

Ah, I see. Well, once you hit threshold you get an action potential so yes in that sense it is either "off" or "on". But a stronger single input is likely to trigger multiple action potentials.

But I'm going to stop being annoying and derailing the Maths thread into brains.

Well, it was interesting — and it is pretty much impossible to derail a math thread, anyway, since absolutely everything is math .

My dad (the former maths teacher) describes it as "the science of patterns". Very poetic.

Not sure whether it counts as a "science" in the strictest sense, though. I mean, is it always empirical? Is it ever empirical? "Art" seems wrong too, though.

I guess in my mind Maths is in it's own division.

Thoughts?

With math, you don't need experimental verification because the theory is rigorous by proofs. So even though it's formal science, it casts doubt on being "science".

It's art because it deals with abstract and astounding things as other liberal arts do.

I read this nice paper on quasi-empiricism in maths by Chaitlin a while ago. Didn't quite agree, but certainly interesting. I forget where it was, though, and in trying to find it I found out that he wrote about something I'm more closely involved in so I'm going to go be distracted now.

But it seems to be a topic he really likes...

edited 12th Jun '11 5:52:11 PM by Tzetze

Re: Turing completeness: Most any computer hardware you name is Turing complete, and is in fact quite a bit better than a 'Turing Machine' per se (which are an epic pain to program). I usually use the term 'Turing complete' of portable devices, as a measure of their programmability (i.e. my calculator is Turing complete, because it's out-of-the-box programmable with real programming languages; an iPhone is of significantly more limited Turing-completeness, since you need to do some fancy stuff, either buying the official SDK or jailbreaking, to code on it at all). Silly usage, I know, but who cares.

I mean, the exponential function is certainly computable, but I doubt that most hardware could compute 2^(2^(2^(2^5000))) or some other Very Big Number of that sort...

edited 13th Jun '11 9:39:20 PM by Carciofus

I never thought I'd say this but I actually enjoy statistics! I find it pretty interesting and I think probability is quite fun too. I've yet to find anyone at my school that agrees with me though; most people were more into trig which I'm lousy at.