Philosophical Discussion of Math

[Photi]

Does anyone have anything cool or intellectual to say about math? I was once pretty good at math up until and including pre-calculus, but then my interests pulled me into the social sciences and I never studied any advanced math after that.

I once took a class entitled "Physics for Social Science Majors." We learned about symmetry, relativity, time and the atom and other interesting topics, but we did so without the use of complex equations, the class was more based on the principles of it all.

Can math be discussed in this way? Math not so much as technique and numbers, but more in terms of a conceptual discussion? I guess I am intending the thread to be open ended, discuss any topic in math not so much with numbers but with words.

For example, a question I have: People talk about Leibniz and Newton inventing calculus, but I wonder, is that really correct? Did they invent calculus, or is calculus a part of nature? Maybe what they invented is merely the symbolic representation of that nature in a comprehensible and useful way? Or, in other words, is math real in the same sense that gravity and light are real? Equations are made to describe energy, mass and gravity and all the other phenomena of the universe, but is this all there is to math, a symbolic expression of natural laws that in their own essence have nothing to do with math? Is there a deeper more esoteric essence to math, an essence that whatever it is contributes to the laws that the universe displays? Why does math work? Is the decimal system for our own convenience, or is there something natural about ten digits? Could a number system be devised based off pi, where pi is a whole number? If so, would this make advanced mathematics any easier or less ambiguous? e the natural number, what's so natural about that number as opposed to any other number?

Don't limit yourselves to those questions, they're there just to get things rolling.

 

[RedRalph]

I think you may have overestimated OT man! I can think in fairly abstract terms, but maths generally go way over my head. From playing the guitar I relaised that more or less everything can be quantified by math, thats about as good as I can do for you buddy

 

[ThinkTank]

Mathematics is about concepts, but about pure concepts, those that are capable of precise definitions, and subject to logic, and logic only. Numbers are an example of that, but there are large parts of mathematics that have nothing to do with numbers.

Now to your question:

Can math be discussed in this way? Math not so much as technique and numbers, but more in terms of a conceptual discussion?

Yes it can but that does not necesarily mean that it will be understandable to a non-professional mathematician. It is sometimes possible to talk about mathematics in a non-technical way that can be understandable for non-experts, as is shown by the class you followed, but sometimes it is just necessary to go into the details.

Having said that, some of the questions you ask make sense and some of them that are in the area of philosophy of mathematics have been topic of discussion for thousands of years. That is especially true of this:

People talk about Leibniz and Newton inventing calculus, but I wonder, is that really correct? Did they invent calculus, or is calculus a part of nature? Maybe what they invented is merely the symbolic representation of that nature in a comprehensible and useful way? Or, in other words, is math real in the same sense that gravity and light are real?

A Platonist would hold that mathematical entities are abstract but real, and that they exist independently of the human mind. So a Platonist would rather say that Leibniz "discovered" calculus (it was already there but just had to be found so to speak). There are various alternative schools of thought on this, for example forms of conceptualism who would say that mathematics is not independent of the human mind, but created by it. According to a conceptualist it would be correct to say that Leibniz "invented" calculus.

 

[Atticus]

Here's some thoughts, starting from the easiest questions:

Can math be discussed in this way? Math not so much as technique and numbers, but more in terms of a conceptual discussion?

It can, and it is in many popular books (such as Infinity and Mind by Rudy Rucker, Gödel, Escher, Bach by Hofstadter &c), there's one trouble though: most of maths requires concrete examples and good explanation. Usually it is easiest to explain maths properly by usin "technical" presentation. If it is not used, reader can have much difficulties to keep the track at the end of discussion.

Any way, conceptual discussion and numbers &techniques do not exclude each other. Even maths is taught at schools as techniques, real maths starts from concepts, symbolic part is there just to make it easier to write, read and comprehend. Just think how much easier it is to write "1+1=2" than "one plus one equals two". When you talk about integrals or more complex things, it eventually comes impossible to remember the start of the sentence when it ends if you don't use any "technical" notation.

Is the decimal system for our own convenience, or is there something natural about ten digits?

Only natural thing with it is that we have ten fingers.

Could a number system be devised based off pi, where pi is a whole number? If so, would this make advanced mathematics any easier or less ambiguous?

You could for example define numbers A,B,C,....,J as A=pi, B=2pi, C=3pi and so on, but if you think it a while, you'll notice that natural numbers 1,2,3,... would be as hard to express as the pi is in the current system. As numbers like 1 and 2 are far more important than pi (pecause they are used for counting), it would only make advanced maths more difficult.

Natural numbers are natural they are learned when we count things, pi is result of little more complex thinking, and that's why most people would have to transform numbers of this new system to the current only to think about them, and after back again to tell what was the results of their thinking.

Moreover, pi isn't such a big problem that people seem to think. Mathematicians just use the symbol and name pi like they use any number, it's nonending decimal presentation is no problem, you just use pi like any number. Actually it's a bit like using the before mentioned "new system" and the traditional system side by side: You can use numers A,B,C,... and 1,2,3,... Only number of the "new system" that is useful just is A, that is pi, because B, C etc are multiples of it.

There are of course other possibilities to cunstruct a pi-based number system, but they all have the same defects as the previously mentioned: natural numbers are poorly expressible and the system is not natural.

e the natural number, what's so natural about that number as opposed to any other number?

As far as I know it isn't called the natural number, you're confusing it with the natural logarithm. Number e is called Napier's constant.

People talk about Leibniz and Newton inventing calculus, but I wonder, is that really correct? Did they invent calculus, or is calculus a part of nature?
---
is this all there is to math, a symbolic expression of natural laws that in their own essence have nothing to do with math?

No, you can do maths without knowing any natural law. The difference between maths and physics is that latter tries to formulate and test natural laws, and they use maths as a tool in their task. Maths on the other hand is about abstract entities, you don't measure circle's circumference but calculate it instead, and the circles of the real world aren't even interesting when you're doing maths. Basically you have the idea of circle in your mind when you're drawing one, and think: "this isn't quite the thing I'm thinking of, but it makes thinking of it easier". Then you try to formulate the idea of circle: "It is a curve which consists of all the points at a certain distance from certain point", and try to deduce from it all the properties you want.

Same way calculus (from the standard point of view) isn't about falling bodies or weight of a stone or something like that, it's abstraction of these problems. If you operate in abstract world, you get obnly abstract results.

Officially maths isn't concerned about the validity of those abstractions, that is: once an abstraction is made, mathematicians examine only it's implications and form propositions of the form: "From these things follows these other things". They don't concern themselves with the validity of their assumptions, only about their implications.

That official interpretation of maths was made around 1900 to secure the back of maths, and it's quite ok. It's only weakness is that while it is highly justifiable, it doesn't tell what maths is to those who practice it (well to most of them at least). Most mathematicians feel they are dealing with something that has existence of it's own, and use the official interpretation about maths only when they are put back against the wall. The question of that existence is a different question, and depends on what you want to call existing.

(Sorry poor and brief formulation of those last paragraphs, the question is quite complex, and I have difficulties of expressing myself precisely in English).

 

[Godwynn]

Does anyone have anything cool or intellectual to say about math?

"If I were again beginning my studies, I would follow the advice of Plato and start with mathematics." - Galileo Galilei

 

[Italian Celtic]

The math is the basis of nature and cuz we are a part of nature we are in part math
the rest is soul

Once every time i listen my mates saying math is useless and only for genius
i would have agreed with them
now i am changed and only now i understand what a big error i made not studing math from basis

if i could give and advice to youngers it'd be " study math study math everytime and everywhere if you wanna be rich and famous"

 

[mdwh]

You could ask the same questions about being real, and existing, of things like music. Does music "really exist"? Does a particular song exist in any sense before a musician writes it?

Personally I think it just depends on how you define "real" or "existence". These concepts certainly exist just like other things such as "music" or "love", but they don't exist in the same sense as physical entities.

 

[Fifty]

philosophy of math is a gigantic field, and there's maybe 4 OTers total that I would think would have any idea on how to give even a semi-sensible answer to any of the questions you posed. A majority of the answers you're going to get here are going to be absolutely terrible.

this article will tell you way more than anyone here can, and give you a bibliography to better sources if you're interested.

 

[mdwh]

* Equations are made to describe energy, mass and gravity and all the other phenomena of the universe, but is this all there is to math, a symbolic expression of natural laws that in their own essence have nothing to do with math?

You can have maths which isn't anything to do with the physical Universe.

* Why does math work?

It works because the Universe has some order to it - things don't happen at random, there are some patterns, and we have developed mathematical systems mainly to model the Universe (e.g., the natural numbers model counting individual items - and the ability to look at a set of non-identical apples and see them as "several apples" is probably hardwired into our brains; real numbers model continuous concepts such as mass).

* Is the decimal system for our own convenience, or is there something natural about ten digits?

Look at your hands.

* Could a number system be devised based off pi, where pi is a whole number?

Depends how you mean - you could have base pi, where pi would be written "10", but then it still wouldn't really be a whole number. Alternatively, you can have non-euclidean geometry where the ratio of the circumference of a circle to its diameter is not equal to the same value that we give pi.

* If so, would this make advanced mathematics any easier or less ambiguous?

I don't see how it would be easier.

* e the natural number, what's so natural about that number as opposed to any other number?

The rate of change of the function y = e^x is equal to y, for all x (i.e., the rate of change of the function is equal to the value of the function).

 

[LightFang]

* e the natural number, what's so natural about that number as opposed to any other number?

The rate of change of the function y = e^x is equal to y, for all x (i.e., the rate of change of the function is equal to the value of the function).

I hadn't thought of it like that! I had just accepted that the derivative of e^x was e^x without really considering the implications! Wow, that's very eye-opening. This is my Aha! moment for the day. Yay for epiphanies.

 

[Bozo Erectus]

I cant speak math, but I can read it and understand it. See my Museum thread if you dont know what I mean.

 

[ParadigmShifter]

As mdwh said, exp(x) is the solution to the question "which function of x is its own derivative?". It just so happens that exp(x+y) = exp(x)exp(y) so it behaves the same as a power function so we can write it as e^x where e = exp(1).

 

[Photi]

philosophy of math is a gigantic field, and there's maybe 4 OTers total that I would think would have any idea on how to give even a semi-sensible answer to any of the questions you posed. A majority of the answers you're going to get here are going to be absolutely terrible.

this article will tell you way more than anyone here can, and give you a bibliography to better sources if you're interested.

thanks for the link fifty! i have skimmed over it enough to know that skimming over this article is not enough. hopefully i will be able to actually read it with concentration sometime with in the next few days. i am sure it will lead to more questions.

as for the explanaton about ten digits matching up to ten fingers offered up by a couple people above, i am not sure i buy that. from that explanation, it is simply a matter of convenience. does our number base have to have an even number of digits? could a number system have seven digits that went something like 1,2,3,...6,10, completely ignoring 7.8.and 9? so 100 would actually be 70, 1000,700 etc. would the ariithmatic still be as easy as in the ten digit system? would all the math still work in whatever field as long as everything was shifted to compensate for the loss of 3 digits? would pi need to be 30% less?

that's actually a good question, what would pi be in a 7 digit system? presumably instead of the fraction 22/7 in a ten digit system, the fraction for pi in a 7 digit system would be 31/10. we know that the division for 31/10 would have to be 3.14..., but then that would mean that in a 7-digit system, dividing by 10 (i.e. 7) would not nearly be as workable as dividing by ten in a ten-digit system. am i looking at this wrong?

 

[Dubai Vol]

I was always good at math (or maths, for you Brits) in school, and it stood me in good stead when I studied engineering. However, I learned that engineers and mathematicians are very different in their view of math(s)

We had a class on "matrix algebra for engineers" where we just learned how to use matrices without going into the theory. We engineers loved that, because that's all we wanted, was to know how to use them as a practical tool. But the grad student teaching the class was clearly distraught, or at least pained, to have to skip the theory. Mathematicians love math regardless of whether it's actually good for anything. We engineers are philistines in their eyes, who just want the tool and fail to appreciate the abstract beauty for its own sake.

My tuppence: whether you think math is an invention of man or inherent in the universe (I vote for invention) study all the math you can in school, whether you like it or not. Everything worthwhile is based on it.

 

[ParadigmShifter]

It doesn't matter what the digits are called. You are just describing a base-7 system. Computer people use base 2 and 16 all the time, because computers use binary and base 16 is easier to deal with than long binary numbers.

You can calculate pi in any base using Taylor Series approximations. There are lots of ways to calculate pi. My favourite is

1 + (1/2)^2 + (1/3)^2 + (1/4)^2 + ... + (1/N)^2 + ... = (pi^2)/6

A more useful one is

atan(x) = x - (x^3)/3 + (x^5)/5 - (x^7)/7 + (x^9)/9... etc.

and atan(1) = pi/4

(atan is the inverse tangent function).

 

[Dubai Vol]

As for pi, it's an irrational number in base ten. 22/7 is not pi, just a close approximation. As far as I know there is not a number system (base 7 or any other base) that makes pi a rational number, meaning one that ends or repeats, like 1/3 is 0.333 repeating for example. You need to consider what pi is: the ratio between the diameter of a circle and its circumference. That ratio is a physical fact, not a mathematical abstraction.

 

[Riffraff]

thanks for the link fifty! i have skimmed over it enough to know that skimming over this article is not enough. hopefully i will be able to actually read it with concentration sometime with in the next few days. i am sure it will lead to more questions.

as for the explanaton about ten digits matching up to ten fingers offered up by a couple people above, i am not sure i buy that. from that explanation, it is simply a matter of convenience. does our number base have to have an even number of digits? could a number system have seven digits that went something like 1,2,3,...6,10, completely ignoring 7.8.and 9? so 100 would actually be 70, 1000,700 etc. would the ariithmatic still be as easy as in the ten digit system? would all the math still work in whatever field as long as everything was shifted to compensate for the loss of 3 digits? would pi need to be 30% less?

that's actually a good question, what would pi be in a 7 digit system? presumably instead of the fraction 22/7 in a ten digit system, the fraction for pi in a 7 digit system would be 31/10. we know that the division for 31/10 would have to be 3.14..., but then that would mean that in a 7-digit system, dividing by 10 (i.e. 7) would not nearly be as workable as dividing by ten in a ten-digit system. am i looking at this wrong?

I think you're overestimating the importance of the number system here. Pi would certainly look diffrent using a different numbersystem - its still the same value though regardless. Other than that - you're correct about aproximating pi as 31/10 in the base 7 system. And also that 31/10 would be the decimal 3.1. You're wrong with the assumption that it would have to equal 3.14 in the base 7 system.

 

[ParadigmShifter]

The ratio of a circle's diameter to its circumference isn't precise enough for mathematicians. We define pi by defining pi/2 to be the smallest positive number x such that

cos(x) = 0 [Note radians are used instead of degrees here]

Crazy but true. We also don't define the trig functions by referring to triangles, they are just solutions to differential equations.

Irrational numbers are irrational in all natural numbered bases by the way. Any fraction that repeats endlessly after a while (such as 123.45454545454545...) is rational (i.e. a fraction). 1/3 = 0.1 in base 3 of course.

 

[Yuri2356]

thanks for the link fifty! i have skimmed over it enough to know that skimming over this article is not enough. hopefully i will be able to actually read it with concentration sometime with in the next few days. i am sure it will lead to more questions.

as for the explanaton about ten digits matching up to ten fingers offered up by a couple people above, i am not sure i buy that. from that explanation, it is simply a matter of convenience. does our number base have to have an even number of digits? could a number system have seven digits that went something like 1,2,3,...6,10, completely ignoring 7.8.and 9? so 100 would actually be 70, 1000,700 etc. would the ariithmatic still be as easy as in the ten digit system? would all the math still work in whatever field as long as everything was shifted to compensate for the loss of 3 digits? would pi need to be 30% less?

that's actually a good question, what would pi be in a 7 digit system? presumably instead of the fraction 22/7 in a ten digit system, the fraction for pi in a 7 digit system would be 31/10. we know that the division for 31/10 would have to be 3.14..., but then that would mean that in a 7-digit system, dividing by 10 (i.e. 7) would not nearly be as workable as dividing by ten in a ten-digit system. am i looking at this wrong?

The base of a number system is entirely arbitrary. You can do the same math in any base you'd like, and still get the same results. Computers use Binary and Hex for all their internal workings, but can easily spit out base-10 numbers for people to work with.

A decimal number in the form (ABCD)n - Where n is the base of the system (and n >= 2) and A - D are all intergers between 0 and n, excluding n - is just short form for D*n^0 + C*n^1 + B*n^2 + A*n^3 ... and so on for as many digits as you'd like. This quantity can then be re-written in any symbolic form we want. The basic arithmatic with the symbols is slightly different (01 + 01 = 10, etc) but all the fancy Math-Wizardry remains the same.

 

[Dubai Vol]

Base 7:
1,2,3,4,5,6,10 etc.

10 (base7) is 7 (base10)=7^1 (seven to the first power)

but 100 (base seven) is 49 (base10)=7^2 (seven squared, or seven raised to the second power)

1000 (base seven) is 343 (base10)=7^3 (seven cubed, or seven to the third power)


Math in any base is just as good as any other, we're just used to base 10. Computer guys use base 16 (hexadecimal) sometimes, because it's simpler in THAT APPLICATION.