Thread: Mathematics

Originally Posted by Wayfaerer

I wouldn't define math as the actual symbols themselves, but the concepts they point to. There are 8 planets revolving around our 1 sun whether we label them as 8 or not. But words don't seem as deeply rooted in nature, the object we call a rock has nothing to do with the word rock whether we label it or not.

Don't take this the wrong way but it doesn't really matter how you personally would define math, its definition is: the study of the measurement, properties, and relationships of quantities and sets, using numbers and symbols. Yes the quantity that 8 symbolizes exists independently of the symbols we use to describe it, I explicitly mentioned it in my last post. I agree.

So if the word r-o-c-k is arbitrary and has no correlation to an actual rock in the world, then what is the correlation between the number 8 and the quantity it symbolizes?

Originally Posted by stormcrow

Don't take this the wrong way but it doesn't really matter how you personally would define math, its definition is: the study of the measurement, properties, and relationships of quantities and sets, using numbers and symbols. Yes the quantity that 8 symbolizes exists independently of the symbols we use to describe it, I explicitly mentioned it in my last post. I agree.

So if the word r-o-c-k is arbitrary and has no correlation to an actual rock in the world, then what is the correlation between the number 8 and the quantity it symbolizes?

What I'm wondering is are "measurements, properties, and relationships of quantities and sets" an intrinsic quality of nature? If so, then math would be an intrinsic quality of nature leave the symbols which are arbitrary.

I'll admit that this is flawed

"There are 8 planets revolving around our 1 sun whether we label them as 8 or not. But words don't seem as deeply rooted in nature, the object we call a rock has nothing to do with the word rock whether we label it or not"

because it is the same exact situation. We do discover distinct entities in nature (which we use words to symbolize) in order to start counting at all.

Originally Posted by Xei

A question I can't answer is why reality outside of human scales seems to be based upon mathematics. I tend to view all human mental constructs as approximations of patterns which will break down when taken outside of the realm in which we formulated them (for instance, our conception of space and geometry as being rectilinear breaks down at scales outside of our experience, yet before knowledge of this many would have claimed that such a conception was 'inherently obvious' and 'an intrinsic part of reality'. Even more fundamental constructs like cause and effect have the same problem).

Yet with mathematics it was totally backwards, in one specific circumstance: we initially came up with complex numbers (two-dimensional numbers involving the square root of minus 1) more than two centuries ago when trying to solve totally prosaic problems (namely cubic equations, which can correspond to real questions about volumes, etc.), but all they were was an intermediary step on the way to the real answer which corresponded to something physical. These numbers turned out to provide a lot of insight into things like polynomials and limits (which were abstracted from intuitive, physical things), even though they themselves were originally an abstraction not corresponding to anything real.

The really weird thing is that quantum mechanics turns out to be pretty much intrinsically based on them. It's all formulated in complex numbers, and the underlying mechanism is based on how these numbers behave; whenever you want an answer, you end up measuring the 'size' of the underlying complex numbers (the size being a normal real number giving a real answer), yet the engine underneath the vehicle is all based on complex stuff.

Why on Earth is it that an obscure academic abstraction that came purely from consideration of macroscopic intuitions ended up being the basis of reality on the most fundamental scales, totally removed from the macroscopic world? How did we find the basis of reality without first having to explore down to this level??

In a way we found a derivative of reality and have been able to further derive and antiderive it .

An interesting essay on the nature of mathematics (invented or discovered):

Mathematical Platonism and its Opposites

I've never liked Math.

Me neither, I see it more as a necessary annoyance for understanding qualitative knowledge of the universe. Other than that I'm more comfortable admiring it from a distance lol.

I find maths to be challenging, highly interesting and most rewarding. You just gotta keep with it, takes effort!

I completely agree with everything you just said, I just was never naturally inclined for the activity.

Originally Posted by Wayfaerer

I completely agree with everything you just said, I just was never naturally inclined for the activity.

Neither was I -so I thought- I graduated high school barely knowing algebra, but I kept studying on my own and trying to catch up to people I wanted to be like, people I read about, or saw on forums, or whatever, and that's how I still go about learning mathematics. I'd say if I really put forth effort, my progress would be around the average sophomore-junior level in college. My pace is kind of slow, considering my age, but I learn better that way. It doesn't matter if you use the basics as motivation or the very tip of the frontiers, it's all fascinating and enjoyable because it everywhere, the best ingredient.

I've always thought that people must be turned off from maths by a lack of self-confidence and poor teaching.

It might look complicated if you've never seen it before but everything we do is literally a sequence of the most retardedly simple, intuitively obvious steps. If something seems hard then you've just not been taught what the little baby steps are properly. I think people seem to think instead that there is some kind of magic ability of insight which they lack, which is the opposite of the truth.

I am almost definitely wrong about this, but hey, you can't fault me for not being egalitarian.

Originally Posted by Xei

If something seems hard then you've just not been taught what the little baby steps are properly.

I've definitely noticed this myself. Sometimes it seems as though most people/textbooks don't reveal the actual simplicity on purpose.

Originally Posted by Xei

I've always thought that people must be turned off from maths by a lack of self-confidence and poor teaching.

It might look complicated if you've never seen it before but everything we do is literally a sequence of the most retardedly simple, intuitively obvious steps. If something seems hard then you've just not been taught what the little baby steps are properly. I think people seem to think instead that there is some kind of magic ability of insight which they lack, which is the opposite of the truth.

I am almost definitely wrong about this, but hey, you can't fault me for not being egalitarian.

Most of the math teachers I had were the ones I disliked the most. They were the biggest assholes out of anyone I learned from. I never struggled with Math. I earned A marks most of the time, but in the more challenging courses, I was a B student. (Pre-Calculus, Calculus) In college, I only took Statistics and Regression Analysis because they were co-requisites with Applied Research Methods which I needed for my my major(s) to analyze data and all that jazz.

I just never found math rewarding, nor pertinent to anything I would be doing in the future. Undoubtedly, I know that it helps with critical thinking and problem solving an immense deal. One of my friends who just became a corporate lawyer told me that getting his extra major in undergraduate school in Mathematics made the LSAT and law school much easier because it opened his mind to other ways to approach difficult tasks and scenarios. I didn't like the process it had (the sequence you mentioned) and I have never been someone to like something so played out. I liked disciplines where I could verbalize, write and communicate my thoughts in various mediums. I never found that in Mathematics. I don't lack confidence in myself in that area, I just don't find it interesting. Things like personal finance, accounting and related things do matter to me.

To each their own though. It is pretty cool to see Math minds go wild on the forums. You guys got good brains.

I'm more so interested in the qualitative aspects of math than actually doing it lol. Not to say I haven't enjoyed using it for physics and other interesting applications, I just personally loose interest toward the skills of the pure activity though they can be awe inspiring. I may learn to like it more...

Originally Posted by Xei

I've always thought that people must be turned off from maths by a lack of self-confidence and poor teaching.

In the UK at least, the GSCE and A level courses always struck me as being somewhat poorly designed. Why are integration and differentiation part of different AS level modules when they're practically the same thing! Not to mention that that level of calculus is so easy that it could be taught years beforehand...


EDIT: Apparently I've developed the ability to send messages 5 hours back in time.

[Double post]

Originally Posted by Photolysis

In the UK at least, the GSCE and A level courses always struck me as being somewhat poorly designed. Why are integration and differentiation part of different AS level modules when they're practically the same thing! Not to mention that that level of calculus is so easy that it could be taught years beforehand...

EDIT: Apparently I've developed the ability to send messages 5 hours back in time.

Several hours, that's pretty impressive bro.

I think they were just in separate modules because of time constraints. I dunno, I think it's good to get to grips with differentiation before you move on to integrals. My main criticism about calculus would actually be that nobody actually has the foggiest idea what an integral is. They can do them symbolically, but if you asked a novel but easy problem, like calculate the length of a curve, nobody would have the faintest idea how it even related to integrals, let alone how to go about the calculation. I don't imagine very many people at all would be able to tell you where differentiation formulae come from, either.

By and large the A-Level is fine, I just think that firstly schools should stream better and not hold bright kids back when they could be doing the A Level years in advance, secondly that there should be less algorithmic questions to test understanding (although that would require competent teachers), and thirdly that there should be a pure A Level, so that kids don't think that mathematicians sit around doing integrals all day, and are better prepared for uni.

Originally Posted by Photolysis

In the UK at least, the GSCE and A level courses always struck me as being somewhat poorly designed. Why are integration and differentiation part of different AS level modules when they're practically the same thing! Not to mention that that level of calculus is so easy that it could be taught years beforehand...

EDIT: Apparently I've developed the ability to send messages 5 hours back in time.

Several hours, that's pretty impressive bro. Edit: I can post before I've even posted though, which is even more impressive.

I think they were just in separate modules because of time constraints. I dunno, I think it's good to get to grips with differentiation before you move on to integrals. My main criticism about calculus would actually be that nobody actually has the foggiest idea what an integral is. They can do them symbolically, but if you asked a novel but easy problem, like calculate the length of a curve, nobody would have the faintest idea how it even related to integrals, let alone how to go about the calculation. I don't imagine very many people at all would be able to tell you where differentiation formulae come from, either.

By and large the A-Level is fine, I just think that firstly schools should stream better and not hold bright kids back when they could be doing the A Level years in advance, secondly that there should be less algorithmic questions to test understanding (although that would require competent teachers), and thirdly that there should be a pure A Level, so that kids don't think that mathematicians sit around doing integrals all day, and are better prepared for uni.

Originally Posted by Xei

Several hours, that's pretty impressive bro. Edit: I can post before I've even posted though, which is even more impressive.

I think they were just in separate modules because of time constraints. I dunno, I think it's good to get to grips with differentiation before you move on to integrals. My main criticism about calculus would actually be that nobody actually has the foggiest idea what an integral is. They can do them symbolically, but if you asked a novel but easy problem, like calculate the length of a curve, nobody would have the faintest idea how it even related to integrals, let alone how to go about the calculation. I don't imagine very many people at all would be able to tell you where differentiation formulae come from, either.

By and large the A-Level is fine, I just think that firstly schools should stream better and not hold bright kids back when they could be doing the A Level years in advance, secondly that there should be less algorithmic questions to test understanding (although that would require competent teachers), and thirdly that there should be a pure A Level, so that kids don't think that mathematicians sit around doing integrals all day, and are better prepared for uni.

Ah yes, I remember my teacher going over where the formulae come from at one point. I think Maths had several proof questions in the various modules as well? Though the questions were often recycled from previous years?

Strangely, at my school Chemistry was the only subject that used streaming at AS/A level, despite most subjects doing this for several years beforehand up to the end of GCSEs. This may have been because it was the most popular science at A level though. Especially compared to Maths, where there were something like 6 people in my class and probably about 15 in the entire year taking the A2 (including Further Maths students).

One thing that was better about Maths and Chemistry (as I recall) was a greater focus on understanding the concepts rather than fact memorisation, despite the limitations like not having a decent understanding of where calculus comes from. Biology was a disjointed mishmash of facts with evolution making a cameo appearance at the final hurdle. Physics seemed to be especially politicized, and while there was quite a bit of calculation if memory serves the questions were fairly predictable and formulistic.

I did like the concept of the AEA papers, and it's something more subjects should move towards.

[EDIT: 2 posts,1 click. sounds like some horrible shock vid. Or the worst title possible for a B movie]

It kinds of depends on your personality type, too. Although I'd argue that mathematics is ideally suited to both ends of the spectrum. Namely, whether you enjoy exploring all details of a very specific niche and like specific detail, or if you prefer coming to understand the very general pictures.

I think I'm at the extreme end of the latter personality type; once I understand the general concepts behind something I will probably become bored of the specifics. I can't stand noodling around with details, to the extent that it can sometimes even makes me physically uncomfortable. And I seem inherently compelled to see things in terms of general structures; this is why I'm engaged with science (the big pictures which explain reality, not little details like the moons of our solar system or specific chemical structures or things), that's why I'm studying mathematics (which is essentially a way of codifying any aspect of physical reality and is really the most general activity possible), and that's why my favourite activity is philosophy.

I imagine you are this kind of person too, but I would guess that you have so far only been exposed to one kind of mathematics, namely the specific kind, for applications (are you well versed in calculus?), which is also the kind of thing you basically do all of the way through school. Now, knowing this mathematics can be pretty cool, in the sense that it vastly increases your competence at manipulating reality (via applied physics), so it's satisfying in the same sense that, say, having the skills to build a suspension bridge is satisfying. And personally I was definitely good at it, although like I said previously, only because I realised it's not intellectually different, and is actually just a tower of very simple elements. But I wouldn't say that hammers or welding are interesting in themselves.

But there is a second half to mathematics, which is that which concerns itself with generalisation, insight, and underlying, overarching structures, which you probably haven't encountered before, which I suppose is roughly that which is called 'pure'.

One example of something studied in pure is group theory. This is the mathematical encoding of symmetry. For instance, a beer mat has a symmetry called D8, which means you can rotate it by right angles, or flip it over. One of the aims of group theory is to examine the structure of these symmetries, and classify them according to a general taxonomy. Group theory can also be applied to physical reality. For instance, Noether's theorem is an extremely powerful and general theorem, which says that whenever a bunch of laws have a symmetry, there will be a corresponding conservation law. One symmetry of the laws of physics is time translation invariance. This turns out to be what's behind conservation of energy.

Another example of something studied in pure is cardinality. The amazing fact is that infinity (the 'number') comes in different flavours; there's actually a hierarchy of infinities. The first is that of the natural numbers, {1, 2, 3, ...}. This turns out to be exactly the same infinity as the number of fractions. However, it is not the same as the infinity of the continuum (the decimals); this infinity turns out to be strictly larger. You can make an even bigger infinity out of that. A natural question is whether there are any infinities inbetween the first and the second. Turns out that this is impossible to prove in standard mathematics, and so must either be assumed to be true or false (which generates two separate branches of mathematical truths). This in turn is directly related to Godel's incompleteness theorem, which states that there exist propositions in any reasonably powerful formal system which cannot be proven by that formal system, unless the formal system is self-contradictory, in which case it is useless.

Does this stuff sound rather more interesting to your sensibilities?