The INTP Approach to Mathematics

[Waterstiller]

This thread is to gather experiences and resources related to math. I'm personally much more interested in hearing about what fascinates and draws you to mathematics. Tell me why you love it and why you're passionate about it; I'd like to catch your passion if you have any.

I want your experiences, what you love about math, and what approach worked best for you in studying it. Also, do you have any recommendations for books or other media? Was learning about the history and philosophy of mathematics helpful for you?

_____________________________________________________________________

In school I always hated math homework and wouldn't do it. But I did well on the tests, so I passed. The way it was taught to me made me averse to math and I've only made it to College Algebra. I don't want this to be a discussion on how math is being taught poorly, but I would appreciate personal histories and frustrations with math (and how you overcame them if you did).

So.. tell me why you love math! Any advice for studying it would be appreciated.

 

[EditorOne]

I enjoyed algebra and quadratics, not that I remember any of it now, but recoiled in horror from an accounting course in college.

Could have done more with mathematics in college except they switched paradigms and started teaching some weird thing called "number sets" just as I went to college. Up to that , through high school, it had been the kind of thing Newton would have recognized and understood, but the "sets" thing was just incomprehensible. So I bailed on math and fed the verbal side of things instead.

The logic of the processes involved in solving equations is innately attractive to me. The tedium of accounting for every single digit in double entry bookkeeping is not. I would rather stick a fork in my eye. (Those using only modern software for keeping accounts may have no idea what I'm talking about. Sorry, I can't rationally explain double-entry bookkeeping except as a system designed by anal retentives to keep obsessive compulsives occupied with meaningless busywork.)

("Bookkeeping" is the only word in English with three sets of double letters in a row. Isn't that interesting? Meaningless, but interesting? )

(Is that little twirling guy symbolic of ducks in thunder?)

 

[Madoness]

As far as I remember, the were two possibilities I was doing... finding out my ways to get to the same answer others were getting.... or writing poetry, somehow in mathematics the surrounding was the best to write it.

 

[RubberDucky451]

As far as I remember, the were two possibilities I was doing... finding out my ways to get to the same answer others were getting.... or writing poetry, somehow in mathematics the surrounding was the best to write it.

I also was interested in finding my own way to do things. I would much rather spend 30 minutes figuring out an alternate route then to wade through 30 minutes of the same problem but with different variations.

 

[Agent Intellect]

I'm actually doing decently in math since going back to college - very different from high school. I think one thing that's helped me is becoming interested in physics, and seeing that math can actually mean something, as oppose to just figuring out what "x" equals or what number to put behind the "=" sign.

Wanting to have a better understanding of physics motivates me to learn the fundamentals of math. Two books that really helped me with that are "Complexity: A Guided Tour" by Melanie Mitchell and "Why Does E=MC^2" by Brian Cox and Jeff Forshaw, but I don't know if those would do anything for anyone else.

 

[B4nd1t_one]

I love math just because its concepts are unarguably true. When applied to the real world (think statistics) math will prove to be a reliable companion in the long run.

 

[AceNess]

Keep on checking every conversion addition and equation and then check it through 5 more times when done. Thats how i do it

 

[Waterstiller]

"Complexity: A Guided Tour" by Melanie Mitchell

Oooh.. that's PERFECT. Many thanks, I'll be purchasing this asap.

 

[Latro]

Doing algebra: I guess a lot. My mind process is "OK, I need to isolate this...what can I do to do that?" a lot of that is automatic, but sometimes it isn't; for instance, I tutored someone the other day who had this question:
Solve for R1.
1/R=(1/R1)+(1/R2)
My first guess was to get a common denominator between R1 and R2, since I had done that so many times in more recent problems, but actually that is not the easiest way to do it, as I found out after solving the problem that way. The easier way is to subtract (1/R2) from both sides and then take the reciprocal of both sides (and then simplify if necessary).

Doing calculus: I guess even more. Derivatives are systematic, but integrals are an art more than a science. I guess various substitutions, I look at possible identities...and eventually if I don't see how to do it I try integration by parts. Either that eventually works (sometimes because it was necessary; other times it just works out eventually anyway but with more work) or that gives me an insight into what I needed to do instead.

Doing things like mathematical logic (proofs): It's a blend of systematic and guessing.

 

[Trebuchet]

I would not have liked math if my dad hadn't taken over my math education. He's an INTP too, and an electrical engineer, and a compulsive teacher. He owned a school, so textbook companies were delighted to send him teachers' editions of any book he wanted, for free. He would order whatever I'd be using the next year, and teach me all summer out of it. This helped, because the books we used were pretty bad. I remember he spent two days trying to figure out chapter 1 of my Trig textbook because it didn't make any sense. In addition to the math, he also taught me how to study it, which is an active rather than a passive process, and confidence in my own ability to learn math.

Since I had this huge advantage, I was good enough at math to have a little fun with it. In Geometry my freshman year, the teacher was oblivious to cheating, so I let the senior in front of me copy my answers. Until the day he saw Return of the Jedi on opening night and told me the next morning that Leia was Luke's sister. Then the jerk had to get his own answers. The same teacher would have someone put the homework on the board, and naturally everyone copied. I was one of about 3 people who ever bothered with the assignments, and I liked to have a bit of fun with them. One day I went to put a proof on the board, and it could have been done in about 4 steps, but I found a cool way to do it in about 30 steps. Everyone copied all 4 chalkboards, and at the end, the teacher pointed out that there was a shorter way. I got some dirty looks.

Pure math has always been entertainment, rather than serious study for me. In college I went for Physics instead of Math. But I have loved writers like Martin Gardner and Raymond Smullyan since I was a kid.

The thing that draws me to math is that there is, in fact, a right answer. English classes weren't like that. The subjectivity of writing Lit papers, and getting grades on what were basically opinions, seemed pointless to me. History and Philosophy and a lot of other things are subjective too, but I liked them better.

Now I have a five-year-old daughter. I try to make math fun, and I treat it like a fun activity. She begs me to give her calculator problems, or to play with her balance, or whatever. Math has mostly brought me joy, despite some bad textbooks and crummy teachers, and I hope for the same for her.

 

[Tunesimah]

Mathematics is interesting for me because it always gives me a good answer to the question 'why'. I crave justifications, nothing can happen just because.... there has to be a reason. Mathematics gives me definitive reasons and justifications more than any other field of study.

I also like breaking subjects down into core simple ideas and building from there. And mathematics is great here too.

Also there's a great deal of creativity and cleverness that goes into solving interesting mathematical problems. This side of math is rarely shown in high school level stuff, and barely even in some college level.

If math was just what I learned in high school, I'd be pretty board with it. There's nothing groundbreaking... everything is just basically obvious. And getting the right answer is less about being brilliant and more about being careful.

In Algebra 2 when I was shown the justification for the quadratic formula... something clicked... and I was impressed at how it was justified. Something apparently out of nowhere could be understood.

When I took Calculus in high school I feel in love. I took a 'reinvent the wheel' approach and tried to justify in my own head the techniques of integral calculus. It's one thing to read it out of a book, it's another to creatively come up with all of it. Once I saw the general idea of Integral Calculus... I closed the book and tried to justify it to myself. Math was and isn't about memorizing formulas for me... it's all about understanding what is going on... why it all fits together.

But it's more than that too, it's also about creativity and play. You have these rules established, what happens if you play with them... what happens if you break them... what happens when you try take a square root of a negative number, what if you establish rules to break what was previously deemed pointless.

I took a look at a pascal's triangle, and I tried to vision it as a 3-D object. What rules pop out when I try that. Is there a pattern? What if I force there to be a pattern.

And ultimately what makes math most interesting is that for all the apparent order and there is a heck of a lot of disorder, or at least what looks like disorder. The prime numbers being a great example. Why are e and pi, such complicated numbers and yet at the basis of tons of mathematics. How can complexity come from such simple ideas?

There's something pure about mathematics that seems impervious to corruption by humanity. It's just truth, plain and simple. And it also seems to chronicle humanities greatest achievements too... these basic concepts are the same ideas that people 1000s of years ago have played around with. It's like being connected to my ancestors... we're thinking about the same exact things.

 

[Berkeley]

I agree with a lot of what you guys are saying. You cannot even be a math person if you just try to memorize formulas. It is intensely satisfying to know the why of something. And like others have said, it is about why this formula works. Everything follows logic. You can't argue with logic. I've had plenty of times where I come up with a mathematical formula for the real world, and every time I do that, either the other person doesn't know what I am talking about, or they take what I say for true. Because that is math, it is truth. You can't argue with truth. I dunno, its beautiful.

 

[Kidege]

Oh! Ephiphany!.

I'm fairly decent at maths, but the last time I enjoyed it was back when I could make the applications tangible. Sadly this was only for a brief period when my mother took the Montessori approach.

So the question was: if I'm good at maths, and I can follow the logic even when I can't see the applicability, why don't I like them more? I just got my answer: because I know, deep down, that if I wanted to create an elaborate and logic set of lies, I could. So maths that don't relate to concrete reality *could* be a lie according to my subconscious.

Thankies Waterstiller.

 

[Latro]

I agree with a lot of what you guys are saying. You cannot even be a math person if you just try to memorize formulas. It is intensely satisfying to know the why of something. And like others have said, it is about why this formula works. Everything follows logic. You can't argue with logic. I've had plenty of times where I come up with a mathematical formula for the real world, and every time I do that, either the other person doesn't know what I am talking about, or they take what I say for true. Because that is math, it is truth. You can't argue with truth. I dunno, its beautiful.

You can't argue with logic, however you can argue with choices of axioms. Even modern mathematics runs into conflicts about how axioms should be set up. There are proofs that work in some axiom sets but not in others, etc.

 

[Ungomma]

I am a mathematician by education, not the best one, surely, but still. Those of you impressed by calculus would have been awed by the beauties of the functional analysis, the theory of measures and general algebra.

I no longer do maths, sadly. And, truth be told, I feel it, feel that it is missing. The intensity of thinking over a problem, the thrill of finally 'getting it' and working towards a solution, the awe and humility its grandeur makes you feel sometimes, the strict logical necessity of its laws and the absolute freedom you have in solving its problems... I spent 5 years studying a subject I will never ever need in my life, but it got me acquainted with the Sublime, for that I'm eternally grateful.

The famous beautiful Mandelbrot set:

The beauty of it is how you get it, though.

Take a complex polynomial: z_{n+1} = z^2_{n} + c; make a series out of it using z_0 = 0.
z_0 = 0 => z_1 = z_0 + c = c => z_2 = z_1^2 + c = c^2 + c => etc.

The Mandelbrot set is a set of all such 'c' that this series is bounded, i.e. |z_n| < infinity for all n.

See? It can be defined in a few lines using high-school math concepts. And the result is infinitely complex.

It produces a feeling of reverence in me. It's like I touched another world with my mind, using these formulae.

A very interesting and passionate article speaks well of what I love about maths.
http://www.maa.org/devlin/devlin_03_08.html
I recommend reading through the first N pages to anyone who respects the Queen of Sciences.

Also, tell me that this isn't beautiful and awe-inspiring:

 

[Aiss]

Regardless of its practical applications, mathematics is a purely theoretical system. You can create mathematical models to describe many things belonging to a physical world, but in itself it's completely abstract. It isn't contaminated with any mundane considerations. It's perfection itself. Shouldn't we just love it?

Rational Portrait of the Architect (INTP)Architects need not be thought of as only interested in drawing blueprints for buildings or roads or bridges. They are the master designers of all kinds of theoretical systems, including school curricula, corporate strategies, and new technologies. For Architects, the world exists primarily to be analyzed, understood, explained - and re-designed. External reality in itself is unimportant, little more than raw material to be organized into structural models. What is important for Architects is that they grasp fundamental principles and natural laws, and that their designs are elegant, that is, efficient and coherent.

I only wish I was actually good at it...

 

[Tyria]

This thread is to gather experiences and resources related to math. I'm personally much more interested in hearing about what fascinates and draws you to mathematics. Tell me why you love it and why you're passionate about it; I'd like to catch your passion if you have any.

I like numbers, and that they make sense. I also like the objectivity that comes with math. While having to follow the rules can be tiresome, it allows for one to reach a logical conclusion (and have others follow as well).

I want your experiences, what you love about math, and what approach worked best for you in studying it. Also, do you have any recommendations for books or other media? Was learning about the history and philosophy of mathematics helpful for you?

I was not a math major, but I took calculus and statistics in college. Looking back, I wish I could retake the courses and put more effort into understanding them. I enjoy creating computer programs, and math (and the mode of thinking) comes in handy. I usually did the assigned problems in the book, and went back to check my answers. I hate it more than anything if a book does not have an answer for a problem, or if I cannot get to the correct answer. At those times, I would usually camp outside of someone's office and get help. I learned a little about the history of mathematics, but I found that the application of math to real world problems was more helpful to me. If I saw a use to what I was doing, I would take more of an interest in it.

_____________________________________________________________________

In school I always hated math homework and wouldn't do it. But I did well on the tests, so I passed. The way it was taught to me made me averse to math and I've only made it to College Algebra. I don't want this to be a discussion on how math is being taught poorly, but I would appreciate personal histories and frustrations with math (and how you overcame them if you did).

So.. tell me why you love math! Any advice for studying it would be appreciated.

My early math teachers did not do a great job with division and other basic mathematical concepts. I struggle when I encounter them (especially complicated equations with x being in the denomenator). As a result, I had to memorize or do simple examples to understand how to do those problems correctly.

I would suggest that you find someone who you can talk to math about, and that their explainations make sense to you (and are correct!) This kind of person will help you out the most when you are studying or lost.

Good luck Waterstiller.

Crim

 

[Waterstiller]

I'm loving the responses to this thread; thank you everyone and keep them coming! This thread is making up for all the shitty math teachers I've had over the years.

It produces a feeling of reverence in me. It's like I touched another world with my mind, using these formulae.

A very interesting and passionate article speaks well of what I love about maths.
http://www.maa.org/devlin/devlin_03_08.html
I recommend reading through the first N pages to anyone who respects the Queen of Sciences.

Also, tell me that this isn't beautiful and awe-inspiring:

That article is incredible. I'm only a couple pages in so far but I'm loving it.

And yes.. that visual is beautiful and awe-inspiring.

EDIT: The article is one of the best I've ever read. Now I'm curious about this "smooth narrative thread that leads from ancient Mesopotamian tablet problems to the high art of the Renaissance algebraists."

I'm thinking about adding his book to my wishlist, but it seems too similar to the article. Hmmm..

 

[Berkeley]

Ya great article, I just finished it.

It makes me feel cheated though.

 

[Waterstiller]

How ironic that math teachers were cheating us out of learning math while their main concern involved making sure we weren't cheating on tests. >.>

 

[warryer]

I'm actually doing decently in math since going back to college - very different from high school. I think one thing that's helped me is becoming interested in physics, and seeing that math can actually mean something, as oppose to just figuring out what "x" equals or what number to put behind the "=" sign.

So true! I didn't have any care for math until I started to learn equations that had actual physical meaning. What?! You mean those numbers can tell me if I push that hard on this metal I can snap it in two!??? /mindblown

 

[Astridian]

My primary area of interest in mathematics is topology. While I have more practice doing point-set topology, the little exposure to algebraic topology I've had has been great fun. I found the first part of Calculus, up to and including multivariable Calculus, to be somewhat tedious, but the subject really started to take off for me when I began studying real analysis. Unfortunately, I have no contact with complex analysis, but I would imagine that I'd take to that as well. I enjoy group theory, ring theory (less than group theory, mind you), and the study of algebras as well. I greatly enjoy using various systems of formal logic for both mathematical and philosophical purpsoes. I love set theory. I regret not getting any exposure to category theory while at the university; I'd imagine that that level of abstraction would suite me well.

My favorite mathematical activity is writing formal proofs. There are few things I feel better about than creating a beautiful proof. There are few things as difficult but at the same time captivating as a challenging proof.

That said, I'm not terribly fond of numbers. That is, while I like mathematics and the study of structures composed of numbers (or of other things), I dislike working with them on an individual basis. I also had a bad experience with statistics at the university, which, combined with some of my philosophical leanings, made me choose to shy away from chance-related subjects somewhat. While I like geometry, I haven't gotten much exposure to it after leaving high school.

My favorite mathematician is Kurt Godel. My second favorite mathematician is Bertrand Russell. I don't have a favorite mathematics book yet, but I tend to admire the books that Dover publishing puts out. I recommend the BBC documentary about Andrew Wiles and Fermat's Last Theorem, to those who have an interest in mathematics.

 

[SEPKA]

I interest in Pure Maths (that is maths minus statistics). Trying to figure out how to prove a theorem most efficiently is fascinating, and I can spend days on it. My teachers helped too because he is the only teacher at my place to actually give out problems on concepts (rather than problems that required you to punch a bunch of calculator button, which I'm bad at because I can't be bothered to find out how to use a calculator). But I think the best part that drawn me in is the making of new mathematical system, and I was once being recognized for it. I'm not sure if people use my project's result, but the delusion that they did motivate me to go on and make more system, even trying to apply these to everyday life. That incompleteness theorem when I find out about it dampen my enthusiatism though, I am still unable to find maths interesting like before.

 

[sniktawekim]

Solve for R1.
1/R=(1/R1)+(1/R2)

if i remember correctly that has something to do with circuitry, maybe resistors in a parallel circuit.. its a shame that im a computer engineer major, and i cant exactly pinpoint that though..

 

[Dormouse]

Reading this has made me conscious of how little mathematical knowledge I have...
I'm still in highschool, though, so I hope that I'll have a chance to learn alot more later on. The level of math were doing seems extremely basic to me, and I know I'm one of the best in my class at applying concepts correctly. ( And literally everything we do is word problems, these days. I'm pretty good at reading/comprehension so I'm at an advantage. I have a few friends who are mathematically inclined but have trouble because of misinterpretation. )
When it comes to finding stuff out for myself, I guess I'm alright. I try, in any case, and some of the most fun I have is devising formulas for random patterns. I'm also interested in topography, and I'm pained that I don't know enough yet to understand it.
I remember in sec one solving a problem by creating a formula for it... It was the wrong thing to do so teacher didn't accept it, unfortunately.

 

[Infinite Regress]

I trade derivatives for a living, and have found the quantitative [mathematical] approach much more satisfying as it provides intuition behind why the derivatives change in relation to other variables [Px, vol, time].

Having said that I'm not a maths junkie, I just want to know enough to make money and leave me with time to pursue my interests. Statistics is bearable for modeling, but stochastic calculus gets on my nerves.

 

[SEPKA]

1/R=1/R1+1/R2 is definitely calculation of effective resistence of parallel ohmic resistor.
One of the time I remember most is when I figure out why a/b =/= c/d if a/b and c/d are both rational number in simplified form and a =/= c. It sounds trivial but is quite a challenge to prove, and when you sit through days and finally solve it, it's fun!

 

[Aiss]

One of the time I remember most is when I figure out why a/b =/= c/d if a/b and c/d are both rational number in simplified form and a =/= c. It sounds trivial but is quite a challenge to prove, and when you sit through days and finally solve it, it's fun!

Either I'm missing something or it's trivial to prove as well...

 

[Latro]

1/R=1/R1+1/R2 is definitely calculation of effective resistence of parallel ohmic resistor.

Similar computations are done in thermodynamics in chemistry.

 

[Razare]

1) Study the concept
2) Practice it once or twice
3) Move on before you get bored

You'll forget it easily this way, but you'll learn it. Only math you regularly use will you end up remembering in the long-run. It all depends on what kind of job you are aiming for.

 

[sotired34]

I love math, but I am horrible at it. I find lectures to be completely unhelpful as well.

I don't know why, but I always tend to look for the core, (usually the most difficult or most important information), and compress it even further into something I can remember, and that's kind of hard to do with math, because most everything is important.

Also, math is something that has to be practiced, and I absolutely hate going back and studying something I have already gone over.

Math is something I want to be good at, as it is a necessity in everything that interest me.