The INTP Love of Math

[Zygomorphic]

A lot of Rationals talk about mathematics with admiration, and I have been trying to get into your minds as to why Rationals like math so much. Ultimately, however, I kept seeing it as "just numbers."

So, today I was taking a multiple-choice exam on reading passages for English and one of the passages particularly interested me (this will thus be going off memory). It was about the mathematician Paul Erdős and his explanation as to why he and other mathematicians appreciate math, and it included an example of what would and would not arouse a mathematician's passion.

Paul Erdős described the aesthetic sentiments toward mathematics, specifically the concept of the proof, as both a science and an art; potentially either trivial or beautiful, and obvious or unexpected. What then makes a proof beautiful and insightful is when the solution is unexpected yet inevitable and economical.

The passage then shifts towards one of historical context, chronicling the Four Color Map Theorem, which many mathematicians are familiar with and had previously accepted as truth. The Four Color Map Theorem was that any type of map (unreal or real) on a flat plane could be filled in with four colors without any bordering regions having the same colors.

So in come the contrarians. One of these skeptics was basically a man who was set out to create a counterexample to this theorem. Every day, during work, this man would whip out a large paper, draw a hypothetical map, and attempt to disprove the theorem to no avail. Eventually, two other mathematicians came to create a counterexample by creating a fundamental pattern of about 1500 maps. Then, they shoved it into a computer, which naturally computed a counterexample to the Four Color Map Theorem. All the of the esoteric, mathematical world was humored; the champagne passed around.

Cool story?

Erdős, although certainly amused that a counterexample was found, perhaps did not find it that fascinating because the process of formulating the counterexample was uninspiring in a sense.

Erdős goes on to say that the capacity for an individual to enjoy the mathematical process is similar to why an individual will or will not appreciate, say, Beethoven's Ninth Symphony; some will innately see the beauty and insight of the work, the rest just see it as a bunch of numbers; a chaotic din.

To be honest, after reading that passage I think I like mathematics - just a little bit...









...but it's still just numbers to me!

 

[Darby]

I am looking into the maths for college, although I think I have decided on physics, and it IS just numbers...to an extent, I think that I see the potential that it has to be something fantastic, although so far I have yet to get that far. i find the manipulation of the numbers to get real things somewhat fascinating, and thats I think why I love math, and it appears to have a limitless font of information and theory in it.

EDIT: I see math much the same way Da Blob describes Subjective and Objective reality(in this case Ideas/Theory and Numbers), they are two representations of the same, they exist within each other, each showing what the other can do.

 

[Trebuchet]

I do love math, though I majored in Physics. I especially enjoy teaching math, and am having a great time watching my 5-year-old daughter play with it.

Why do I love it? I've tried to answer that many times, usually for people who asked a question much like this. First, yes, there is beauty to it. Many people who are much smarter than I am have shown this in ways that anyone could appreciate, I think.

Antony Garrett Lisi, a theoretical physicist, was working on a unified field theory, aka theory of everything, and came up with a mathematical model that is usually represented as a beautiful graph called E8.

The Mandelbrot set and Julia sets are obviously beautiful, which is why they show up on tee shirts. Check out this page of beauty.

Penrose and Escher both used geometries to create beauty.

There is more subtle beauty, such as the appearance of Fibonacci numbers or the golden mean in nature. Or prime numbers, or casting out 9s, or figuring out probabilities, determining a correlation, or seeing the relationship between triangles and circles in trigonometry.

Math is also fun. It can be a great pastime, an entertaining way to challenge yourself, and make discoveries that are at least new to you. Martin Gardener was one of several writers who excelled in showing his readers how to play with math, like a toy. Teaching math is fun, because you get to watch someone else discover it.

And math is comforting. There is a right answer! Maybe we don't know what it is yet, but there is one. This is not at all true of, say, history, something else I enjoy but for totally different reasons.

I don't do math professionally, and I don't think it would have been as good a match for me as Physics. These days, I spend my time being a mommy, mostly. But the love remains.

Most people I know who hate math were taught badly. It was a rote, boring, pointless subject. I was taught by my dad, and I had a few wonderful teachers along the way, and I never doubted the fun of it, or my ability to master it, even in the most poorly taught class. (For example, in 7th grade, my math teacher told me not to bother because girls aren't good at math. He made people do pushups if they answered wrong. The class wasn't fun, but the subject was.) So my love of math is also based on luck.

Oddly, I don't really love that most mathematical of composers, J. S. Bach. Oh well.

 

[Agent Intellect]

I think one of the most fascinating things about math is that our universe works in a way that it can be described mathematically - it doesn't follow arbitrary laws, or laws that are changing from moment to moment. Math is universal, it's relatively simple (if you can remember the little rules), and the more I do it, the more I keep thinking about things in mathematical terms (coming up with formulas in my head to explain things I come across during the day). Some physicists have described math as having like a sixth sense, since so much of it can really only be understood mathematically (like n dimensional space) - I loath the idea that I may be missing out on an integral part of reality by being mathematically blind.

 

[Cognisant]

Algorithmic processes fascinate me, it's beautiful how a seemingly simple series of symbols can represent a mathematical process which in turn makes software behave seemingly intelligently.

It's a language of patterns.
And who doesn’t love finding/playing-with patterns?

 

[morricone]

...but it's still just numbers to me!

Then you never saw a real mathematican at work.

"Physics is the study of the world, while mathematics is the study of all possible worlds."

"Mathematicians are artists without an audience. Everyone can appreciate a musician playing a musical piece, but to appreciate the beauty of a mathematical proof, you have to be familiar with them."

Paul Erdős often spoke about proofs from "The Book" in which God keeps the most elegant proof of each mathematical theorem.

I'd wish I could recommend Proofs from THE BOOK. But yet I still have to get it myself.

 

[Aiss]

The passage then shifts towards one of historical context, chronicling the Four Color Map Theorem, which many mathematicians are familiar with and had previously accepted as truth. The Four Color Map Theorem was that any type of map (unreal or real) on a flat plane could be filled in with four colors without any bordering regions having the same colors.

So in come the contrarians. One of these skeptics was basically a man who was set out to create a counterexample to this theorem. Every day, during work, this man would whip out a large paper, draw a hypothetical map, and attempt to disprove the theorem to no avail. Eventually, two other mathematicians came to create a counterexample by creating a fundamental pattern of about 1500 maps. Then, they shoved it into a computer, which naturally computed a counterexample to the Four Color Map Theorem. All the of the esoteric, mathematical world was humored; the champagne passed around.

Cool story?

Erdős, although certainly amused that a counterexample was found, perhaps did not find it that fascinating because the process of formulating the counterexample was uninspiring in a sense.

I hope none of the questions was about whether a theory was proved or not, because you seem to have misunderstood this part. From Wikipedia:

A number of false proofs and false counterexamples have appeared since the first statement of the four color theorem in 1852. (...) The four color theorem was proved in 1976 by Kenneth Appel and Wolfgang Haken. It was the first major theorem to be proved using a computer.

 

[Zygomorphic]

"...thus proving that a minimal counterexample to the four-color conjecture could not exist."

That was from the same Wikipedia page. Perhaps the passage was misread due to it being worded confusingly and similarly to the above quote (i.e. proving that it is not counter-able).

Fortunately, the point of the first point still stands.

Unfortunately, I probably did get a question wrong.

 

[kantor1003]

As x-cognistant pointed out, it is a language. I guess that's why math appeal to me on some level. It is a more objective/concrete language then the one we use in our daily lives. Explaining theory in language can be a painful event because the message is so easily distorted, counter argued and it can often turn into a semantical debate. Needless to say, it is a lot of hassle. A mathematician can just show his theory mathematically and to other mathematicians it provides a clearer image then the linguistic approach.

I am not into math, so I don't know that what I say holds true on any level. It is just my current romantic view of math. If it is "wrong", please correct me.

Wish I had an early interest in math and became proficient with it. School managed to destroy any hint of potential interest in it.. damn school.

 

[Achilles]

I'm a philosophy student personally, but my main area of interest is in logic and the foundations of mathematics (and I take a great, albeit negative, interest in set theory - set theory is evil) and I can give a couple of good reasons to like maths for its own sake:

Firstly, and most importantly, it's beautiful. Giving any more description than that is like trying to give any more explanation than "It's beautiful" to why you like your favourite poem. (And poetry really is the best comparison, although some might say music) If you don't get why it's beautiful then you probably never will - but I can assure you you're missing out on something, just as much as the philistine who refuses to read Blake (or Yeats, or Auden, or [insert your favourite poet here]) would be.

Secondly, it allows for certainty. I personally think Certainty to be one of the most fascinating phenomenon and concepts around (for, it is both - it is a psychological fact about people that they sometimes "feel" certain, and act upon this, and we also have a concept of certainty, evidently, which lets us make such ascriptions) and mathematics is one of the best case studies around for it.

Also, for people more interested in explaining the natural world, maths is very good for that too. But I consider natural scientists to be barbarians so, that's not for me!

 

[fullerene]

Antony Garrett Lisi, a theoretical physicist, was working on a unified field theory, aka theory of everything, and came up with a mathematical model that is usually represented as a beautiful graph called E8.

That sent a shiver down my spine. The Circle has been the fundamental understanding for reality since... since... well, since as old as the oldest texts that I've read. Euclid actually included "being able to draw a circle with any center and any radius" among his 5 fundamental postulates, in his Elements, and if you look at the propositions (things he proved using those postulates), they lean much more heavily on the circle than they do on the line.

Then of course there's the simple observation that for the longest time, people thought that the stars and planets revolved around us circularly, rather than elliptically, even to the point that they hypothesized epicycles, spheres attached to spheres, in order to explain why observations didn't line up with the perfect, concentrically-spherical universe that they first hypothesized. And after that, epicycles attached to epicycles (same link).

Correct me if I'm wrong, but I think an infinite sequence of epicycles rotating reduces, mathematically, to an ellipse? They were just working with the first few terms of an approximate expansion, rather than the analytic infinite sum that elipses prove to be.


I don't get surprised by that many things in life (anymore), but the continual emergence of circles whenever a description of reality is concerned. And of course the circle is the base behind pi (ratio of the circumference to diameter of any circle), which pops up in the ways which are normally considered "beautiful math". And, though I couldn't understand it well at all, this site seems to suggest that e and pi are inseparable?

I love the circle, which is, imo, probably the shallowest-representation of the shape of reality--like a cheap imitation of whatever the real shape (probably some multi-dimensional spiral-like thing, being drawn out with a time dependence as well) is. But it all seems to begin with this shape... which, since no "real" (that is, physical and perfect) circles exist where our senses can reach out and touch them, has to be studied in our minds, using math, instead. I also think that this is the segment of reality that science misses out on, simply because it doesn't lie within the scope of repeatably testable. It's really a shame, because science uses the things that math proves constantly (I don't really think of what we do in physics "math" anymore), but most people immediatley get preoccupied with the science and completely ignore the truths that the math tries to make apparent .

I'm studying to be one of those barbarians achilles spoke of , but I agree with him completely. (edit in here: I actually didn't even see that he said he was a philosophy student before I wrote this... but it fit so well!) Everyone in the philosophy/math -> natural science -> engineering -> repairman chain can pretty easily look down on the level below them and think "they're butchering my subject!" Repairmen are appreciated because they make things work, but after something is created, the engineer is forgotten. Engineers are appreciated too, because they make new things... but scientists are less so. People are a little bit more sceptical when it comes to natural scientists. There you're pushing out of the range of "practical" for an everday person, so they start getting that "oh, it's cool for academic purposes... but we've pretty much gotten all we need from it" bias that some people start to care less about. Natural scientists are the cusp; in front of them, work is appreciated as "real" and "useful", they themselves are partially appreciated and partially neglected, but behind them there is almost wholly neglect. Mathematicians are appreciated for.... their ability to break encryption codes in wartime? If the public even recognizes that they're useful for that particular skill? And philosophers are even less appreciated than that. I, personally, would also put theology -> philosophy at the top of that chain too... but theology is so long forgotten that I'm almost obligated to leave it off in order to be taken seriously.

It's all basically the same... another face to the problem that people care more about shallow things than deep ones, hold physical things more highly than nonphysical ones, and otherwise want results now. I appreciate math for no other reason (or at least, no other cause) than because I was lucky enough to be taught by people who appreciate it, and some of that appreciation rubbed off on me. It is definitely well worth the time to study, though, if you have the ability. I just wouldn't recommend doing it in most schools.

 

[Achilles]

I'm studying to be one of those barbarians achilles spoke of , but I agree with him completely. (edit in here: I actually didn't even see that he said he was a philosophy student before I wrote this... but it fit so well!) Everyone in the philosophy/math -> natural science -> engineering -> repairman chain can pretty easily look down on the level below them and think "they're butchering my subject!" Repairmen are appreciated because they make things work, but after something is created, the engineer is forgotten. Engineers are appreciated too, because they make new things... but scientists are less so. People are a little bit more sceptical when it comes to natural scientists. There you're pushing out of the range of "practical" for an everday person, so they start getting that "oh, it's cool for academic purposes... but we've pretty much gotten all we need from it" bias that some people start to care less about. Natural scientists are the cusp; in front of them, work is appreciated as "real" and "useful", they themselves are partially appreciated and partially neglected, but behind them there is almost wholly neglect. Mathematicians are appreciated for.... their ability to break encryption codes in wartime? If the public even recognizes that they're useful for that particular skill? And philosophers are even less appreciated than that. I, personally, would also put theology -> philosophy at the top of that chain too... but theology is so long forgotten that I'm almost obligated to leave it off in order to be taken seriously.

Ah, the barbarians comment was all in good spirit!

And, have you seen this? http://xkcd.com/435/ Your comment reminded me of it! Me and my fellow logic enthusiasts (all either philosophers or mathematicians or computer scientists) were all deeply annoyed at the fact that it's not a logician standing behind the mathematician saying *ahem*.

But, yeah it's true what you say in the bit I highlight, and I engage in a bit of that myself. But, really, everybody has their place to play in the world and all and it's important to recognise that. However, I can't help but feel that mathematics captures something far more beautiful than is ever found in any natural science. Nature is just something we bumped into while we were looking for something to kill or have sex with, mathematics is something we have found only be conscious effort (either by invention or discovery, not wanting to get into that debate now) and honed and perfected with great tenacity. It can't be denied there is beauty in nature (not by anybody who has seen a rainbow) but I think mathematics still has an edge over the natural sciences because the beauty we find there is a result of human effort, and is the culmination of striving and achievement against the odds - another thing it shares with the arts.

I also like what you say about theology, and (though I have no personal interest in it) suspect it's true; theology is the purest subject of all, and the deepest. Unfortunately, however, as you suggest, there is a bias against theology in this age.

 

[fullerene]

hehe sure, everything's meant in good spirit when someone else shows respect for your discipline . I was going to be a physics/philosophy double major, and actually took several logic courses, but stopped after I suffered through a class called "nature of reason" with a frustrating professor. If majoring in philosophy, a subject whose value (to me) lies largely in critical thinking skills and wrestling with vaguely-defined questions, was going to make me sit through classes where we had to regurgitate what different famous philosophers thought "reason" should be without processing them at all, I really didn't want to deal with it.

I have seen the xkcd. And I mean... I guess the things are wrapped together, in a much more complicated way than my linear-progression made them seem. After all, neuroscience is also just applied bio/chemistry, but it may someday provide a foundation for logic. Or it may yank the foundation out from under logic, by showing that it's a product of how the human mind is wired--and then logic just becomes applied neuroscience, and we're left with no foundation at all (except for "what is somehow told/input into your mind by a somehow-infallible source," I suppose, which I think is still theological in nature... although I doubt a vaguer subject for study ever existed, haha).

I agree, too, that math and the arts are around the same level. I'm not sure it's because the beauty you find is the result of human effort, though. I mean, have the natural sciences risen apart from human effort, and is the only beauty in them found by someone simply looking at the world? I mean... so for example, we know, both physically and mathematically, that every electric charge distribution acts like a point charge from far away. This didn't come without lots of study in physics... we needed to know that charged objects exist (the "looking at the rainbow" level of things), but also how they acted (coulomb's law, or maxwell's equations... whichever you prefer). Then we also needed to know how to extend finite point charges mathematically into continuous distributions (integrals/calculus), as well as take limiting cases where the observation point is far away, compared to the dimensions of the system (taylor series expansions). All this effort just to know that any charge distribution will cause other charges to "see" it as a point charge, from far away.

The beauty in this is that people are much the same way. Stereotypes require people who are distant from a group to view the people in that group as the same. Ideas that are quite distinct to you, as a logician (boolean algebra as opposed to Heyting algebra, to grab the first two different-but-similar logic systems I could find from the wikipedia page on propositional calculus ) are just "logic" to other people. The beauty comes from the revelation that you can learn something about yourself and how you ought to evaluate ideas based on electric charge interactions.

This revelation did not come without considerable human effort in the natural sciences... but at the same time, do I think it's really any more beautiful than what math turns up? Honestly? Not really. I'm pretty sure that there's something else that math and art trigger apart from the fact that they take effort, which supercedes the beauty that's to be found in the natural sciences... or at least in physics. I don't really know what it is, but I really think that it's to society's detriment that it doesn't care anymore.


welcome to the forum, by the way.

 

[Achilles]

Jesus I hate most of my philosophy modules. I have a policy of just taking everything logic related (various logic modules, philosophy of logic, philosophy of language, foundations of mathematics) from the philosophy department then doing either maths or psychology modules to make up the rest. However, unfortunately, I have some "core" modules which I have to do as part of a philosophy course. So I must endure a lot of ill defined and confused thinking which leads to nowhere.

That being said, I very much enjoy some philosophy which isn't so rigorous as all that. If you know anything about the ordinary language school (and Moore and later Wittgenstein) I'll just note that I very much enjoy the stuff they do. My favourite philosophical proof is now and always will be Moore's proof that time is in fact real. It went something like: It is said by some philosophers that time is an illusion. But that can't be right, because today I had a bath, then I went for a walk, and after that I had a nice cup of tea. If you're going to stay at the level of common sense and intuition at least do it properly, like that!

After all, neuroscience is also just applied bio/chemistry, but it may someday provide a foundation for logic. Or it may yank the foundation out from under logic, by showing that it's a product of how the human mind is wired--and then logic just becomes applied neuroscience, and we're left with no foundation at all

I'm not sure (and I mean that in the sense of genuine uncertainty) whether that is a possibility or not. Imagine the paper presented such and such evidence, then gave such and such arguments, and then said in light of the evidence combined with this argument, we would best conclude that logic fails to be reliable. But, then (and I have phrased this description to make it obvious) why would we believe the conclusion in light of the evidence? There are probably ways around it but it would depend on the details of the account, and I confess I'd be sceptical of it. Sometimes people say very radical things indeed just because it gets ones paper attention. Plus, I don't think I could bare all the post modernists (who I already get flack from as a logician) harping on about what a triumph that would represent for them.

That's a very good point about natural science having a similar sort of beauty. I shall have to rethink my position. Maybe it's just my own personal prejudice: I happen to prefer the formal sciences, and some of philosophy, and the literary and musical arts, so I have come up with a rationale for doing so.

And thanks for the welcome, the pleasure is all mine

 

[lab_tech]

me?
ITNP

But?
hate math

Why?
prefer to value my time more philosophers thought than solving math calculation....not all, everyday issues cant be solved by math.

 

[lab_tech]

But?
Dont hate those mathematician, love em....darn fukin genius

 

[Starfruit M.E.]

I like to find connections. For instance, I could use a derivative (a mathematical concept I learned in Calc 1) to find the velocity or acceleration function when I only stared off with a position function. And that connection was really exciting to me. I also like Physics because of the connections and answers I found there. But I honestly hated algebra until I figured out all of these connections. I think you really do have to be willing to stick with math until calculus to be able to appreciate what is happening, though, otherwise you never get to the good stuff.

 

[fullerene]

I thought that too, at first... but then I found that discrete math can actually be really cool. You just don't usually get to any of the cool discrete until post-calculus (or at least, to math around that same level). The pre-calc stuff is just usually taught in high school and below, which is a shitstorm as it is.

Even algebra is somewhat cool, if you keep in mind what you're doing. You're saying that given an equation relating various quantities, you can be sure that you preserve truth if you stay within the rules. Algebra is like the study of a perfect manipualation of knowledge, that if you know a certain relationship exists, you can find out infinitely many more true relationships by manipulating them carefully. It's a very crisp and concrete image of pure logic in action. Like logic, you can't just "use algebra" to get anywhere--but given a starting point, the possibilities are endless.

(I'm really glad achilles and I didn't kill the thread, by the way. I really wanted to see what other people had to say. I'm right on that cusp of understanding with math where I appreciate it and can follow what people say, but if you bring up any cool relationships that you especially appreciate the odds are that I haven't heard them yet. It's the most exciting/easy time to learn about something).

 

[givalentine]

I have never been interested in math unless there is a puzzle to solve, unless an equation carries greater meaning outside of the context of a book or problem. Neither numbers nor patterns really interest me for this reason. I have always been interested in philosophy instead, but I am quite good at math anyway. The only other reason I would enjoy it would be it takes my mind off of other thought processes and concentrates my effort in one place.
The best possible way I can put it is that math thus far, has only provided solutions with little purpose, whereas philosophy is all purpose and hardly any solutions.
For me, solutions, facts, and truths are of no interest (they simply do not exist to me)unless I can use them to serve a greater purpose or find meaning in something else. The abstract world does not affect me unless its principles can provide explanations for real-life scenarios or work as evidence to prove something that affects my life.
It would be my expectation that a majority of INTPs also think this way, and mathematics provides them with greater purpose. I would think that is why INTPs likely become engineers and architects - applying their interest to make something rather than working with math only on an abstract level. I suppose it depends on your experiences with math. Mostly, how you were introduced to it and whether it initially served its purpose and provided answers that adequately aided you in life. If it sparked interest and an INTP realized it could gain the needed knowledge, they likely stuck to it. Otherwise, they were driven toward the more philosophical and psychological subjects. Or perhaps, both compete for interest because they are both logic-based sources of knowledge.

 

[Adamastor]

Ahhh... I agree with so many things that were said.

I wouldn't go as far as to state that I love math, but I like it.
Math is beautiful and, even though people describes the reasons of why math is beauty, I kinda of like this inefable point of math, so I try not to bother with it

I mean... Its is common for anyone who truly enjoys math to spend a great deal of time playing with it. Who hasn't spend a few hours dealing with a problem, just to get frustated and look at the answer and then laugh about it?

Personally I find Calculus dull and boring and I am certain that there many people who shares this thought. What I like about math (and I am not sure if this "belongs" to math at all) is the simple situation to attack a problem, creating, hopefully a solution. To be successful one has to own the right tools, as to know how and when engage and things like that...

Take that for instance:

Even algebra is somewhat cool, if you keep in mind what you're doing. You're saying that given an equation relating various quantities, you can be sure that you preserve truth if you stay within the rules. Algebra is like the study of a perfect manipualation of knowledge, that if you know a certain relationship exists, you can find out infinitely many more true relationships by manipulating them carefully. It's a very crisp and concrete image of pure logic in action. Like logic, you can't just "use algebra" to get anywhere--but given a starting point, the possibilities are endless.

Its not like when I was learning algebra I was ecstasy or anything similar... The algebra itself is not a big thing: playing with random equation might not amuse everyone, but playing with a equation you had come up with, after analysing a situation, the abstraction of a situation, got a total different meaning! As cryptonia said, if you preserve the truth you'll discover new things about what you are dealing with, right? (That was the whole purpose right?)

__

...but it's still just numbers to me!

I like numbers!
Numbers are fascinating, It has always amazed me the power of abstraction of numbers. Number theory is my favorite math's branch because its so much fun putting all things together (well, we are dealing with something elementary, fundamental, are we not?), discovering new things, this feelings I suppose it is similar to the feeling of an archeologist's findings.

I think it would be cool to add, that computer come really handy at these times. I'm not a calculator like gauss was (to do astronomical divisions with only my head), but doing the computer displaying the series you are working with in a nice way as to draw conclusions, looking at the patterns is more human I believe...