Is Mathematics the Best Language for Understanding the Universe?

[WindScars]

I'm very good at math and scored high during all my life - but I must admit that nothing I have found until now couldn't be done better by programming. So, if someone wants to understand the nature of the world, not for an university or for jobs, but for it's sake - can you provide me one single example of where advanced math could be more useful than just understanding the subject and using computers to do the calcs.

This topic is not very clear, so rephrasing:

It's about changing arcaic hand algebra to programming as the main tool for humans to work with math. Where I wrote "math skills", visualize it as youself working into your algebra on a paper, getting an integral algebrically, solving a differencial equation by hands. Couldn't this be replaced by understanding what an integral is, and using the computer to solve it for you? You don't have to get deep into every topic to use it properly. Time wasted is advancement lost. I would like to see one example where all that time spent on: decoring formulas, getting algebra skills, learning to solve an integral alebrically, and others, will be more useful than doing it programatically.

Thank you.

 

[DivisionByZro]

Cryptography. Say I give you an integer with around 180 digits. Now I ask you to find a prime factorization for that number, what do you do then? Brute force? It's a very tough problem in general, and you WILL need advanced math to even attempt it, let alone make a program.

 

[TMFKAN64]

I think that the problem is more one of checking the computer's accuracy. Numerical methods can be unstable for certain classes of problems, and all non-trivial programs have bugs anyway.

If you don't understand the math, you won't be in a position to properly evaluate a computer generated answer.

Just think about calculators for a second. If I ask you to divide 117.938 by 19.767, clearly a calculator will be much quicker. But if you come up with an answer of 56.556, how will you know that you accidently pressed '1' twice without knowing that 120 / 20 is about 6?

 

[Haborix]

First, programs, to be of any use, must be written by someone who understands the mathematics that goes into doing any calculations.

can you provide me one single example of where advanced math could be more useful than just understanding the subject and using computers to do the calcs?

I think you make an unwarranted dichotomy between advanced mathematics and understanding a subject. You can claim to understand physics without math, but I would argue it's through a set of vague analogies. Mathematics gives structure and definition to analogies and allows you to know in what ways the analogy is correct or incorrect.

I personally believe that everything we do in physics is model building that has no real truth in nature. Consider an artist that makes a copy of a painting. You would argue that visually it is a similar painting, but it is never the original. Going back to physics I think learning physics without math is like making a photocopy of the copy of the original.

Babbling done.

-Eric

 

[micromass]

can you provide me one single example of where advanced math could be more useful than just understanding the subject

Advanced math is needed to understand the subject. Try to understand quantum mechanics without knowing linear algebra. How will you understand http://en.wikipedia.org/wiki/Schrödinger_equation without knowledge of PDE's??
You can't just separate advanced math and physics.

Furthermore, who will write all this fancy programs for you?? People who know the math!

 

[WindScars]

TMFKAN64 you don't get the point. The computer can provide wrong answers as you. This has nothing to do with the issue. You can check it. I'm proposing a replacement of hand algebra to programming as the primordial tool to your self development of scientific knowledge. Literary: changing the paper to the keyboard.

Same to Micromass. It's about changing arcaic hand algebra to programming as the main tool for humans to work with math. Where I wrote "math skills", visualize it as youself working into your algebra on a paper, getting an integral algebrically, solving a differencial equation by hands. Couldn't this be replaced by understanding what an integral is, and using the computer to solve it for you? You don't have to get deep into every topic to use it properly. Time wasted is advancement lost.

Reading the topic again I admit it's not clear so this is bad news for the proposed discussion.

 

[micromass]

TMFKAN64 you don't get the point. I'm not saying in not understanding the subject and hoping the computer will do everything for you, that's the case when you ask it to do a division without knowing what to expect. I'm proposing a replacement of hand algebra to programming as the primordial tool to your self development of scientific knowledge.

A respectable mathematician (don't know his name, sorry), went to an elementary school where they focussed on abstract math instead of doing algebra. So he went to one of the kids and he asked him: "what is 2+4". The kid thought for a while and answered "It equals 4+2 because of commutativity". The kid had no idea what 2+4 meant and that it equaled 6.

I'm sorry, but I find this very sad. People need to know how to calculate. They need not be good at it or it needs not be fast, but they need to know how it goes. You can't possibly talk about integrals and derivatives without ever having calculated one.

 

[DivisionByZro]

TMFKAN64 you don't get the point. The computer can provide wrong answers as you. This has nothing to do with the issue. You can check it. I'm proposing a replacement of hand algebra to programming as the primordial tool to your self development of scientific knowledge. Literary: changing the paper to the keyboard.

Same to Micromass. It's about changing arcaic hand algebra to programming as the main tool for humans to work with math. Where I wrote "math skills", visualize it as youself working into your algebra on a paper, getting an integral algebrically, solving a differencial equation by hands. Couldn't this be replaced by understanding what an integral is, and using the computer to solve it for you? You don't have to get deep into every topic to use it properly. Time wasted is advancement lost.

Reading the topic again I admit it's not clear so this is bad news for the proposed discussion.

You clearly have never done any higher math. What you are basically saying is: "Let's get an idea of what a car is, and then let's forget about building one ourselves, let the robots do it.", except that you almost make it seem as if that's equivalent to spending the time building the car yourself.

How would you ever develop any mathematical maturity whatsoever? How would you advance mathematics if you didn't develop intuition and problem solving skills? Mathematics is not just a "read it, know it" type of monster. It's a common saying that you have to fight math to understand math.

 

[WindScars]

A respectable mathematician (don't know his name, sorry), went to an elementary school where they focussed on abstract math instead of doing algebra. So he went to one of the kids and he asked him: "what is 2+4". The kid thought for a while and answered "It equals 4+2 because of commutativity". The kid had no idea what 2+4 meant and that it equaled 6.

I'm sorry, but I find this very sad. People need to know how to calculate. They need not be good at it or it needs not be fast, but they need to know how it goes. You can't possibly talk about integrals and derivatives without ever having calculated one.

Yes, your example is perfect. Why is being able to calculate an integral more important to knowing what it is and what it does? If anything, understanding rules as d/dx x^n = nx^(n-1) is essentially what happened to the 2+4 guy.

You clearly have never done any higher math. What you are basically saying is: "Let's get an idea of what a car is, and then let's forget about building one ourselves, let the robots do it.", except that you almost make it seem as if that's equivalent to spending the time building the car yourself.

How would you ever develop any mathematical maturity whatsoever? How would you advance mathematics if you didn't develop intuition and problem solving skills? Mathematics is not just a "read it, know it" type of monster. It's a common saying that you have to fight math to understand math.

You clearly have never done any higher guessing. Neither reading. Quote where I said against fighting math, developing intuition and solving problems. I'm arguing in favor to using programming as a replacement of the old pen and paper. You clearly have no idea of what I'm talking about.
People are strange. I'm done here. Got my answer. Thank you.

 

[micromass]

Yes, your example is perfect. Why is being able to calculate an integral more important to knowing what it is and what it does? If anything, understanding rules as d/dx x^n = nx^(n-1) is essentially what happened to the 2+4 guy.

Personally, I want to feel comfortable with the material. And to be comfortable, that means that I absolutely must see and compute actual examples. I have tried reading through a math text before without doing any computations. I remembered very little and I was not comfortable with the material at all.

Just knowing what an integral means is ok. But never calculate one makes you out of touch with what those things actually are. It's hard to explain, but it's really true.

Furthermore, what are you going to do if there are no numbers anymore, but only symbols?? How will you derive a formula from a given law?? If you don't know how to handle the numbers, then you will be hopeless if there are symbols instead.

 

[DivisionByZro]

Well I suppose you and I have different views of what doing math on a computer entails. I did not mean to offend anyone, perhaps I was unintentionally harsh.. There's a good reason pencil and paper is a tried-true-tested technique for math...

 

[Office_Shredder]

The military wants to write an algorithm to track am object falling out of the sky. You solve the differential equation numerically every timed you want to update the position of the object with your best guess, therussian programmer solves it once and just repeatedly evaluates the solution. Your program takes too long to run to successfully track the enemy nukes and the russians win the cold war in a very hot mess

 

[WindScars]

Don't you think weird a technology available for thousand years is still the one used here? They didn't use ink and paper because they had a choice, this was just the only way they had to keep data and express concepts. Programming is essentially the same. There's nothing I can express on a paper I can't on notepad - except on the later I replace stuff like dividing by hand, solving an equation, and others, by an automated method. I won't lose a single bit of the knowledge. When you are solving an equation or performing a multiplication you are not thinking on your main problem anyway, it actually takes you off the focus. You know it.

Now this last answer really did it for me. I got an insight here. Thank you guys.

 

[micromass]

You clearly have never done any higher guessing. Neither reading. Quote where I said against fighting math, developing intuition and solving problems. I'm arguing in favor to using programming as a replacement of the old pen and paper. You clearly have no idea of what I'm talking about.



People are strange. I'm done here. Got my answer. Thank you.

Oh come on. You asked a question, Div gave a fairly good answer. If you disagree with him, then bring in counterarguments. The thing you mention is very interesting, and I would love having a discussion with you about this. Don't back off because people disagree with you.

Perhaps Div misunderstood you, well: try to explain better to him what you mean.

A very related issue is whether mathematical proofs are actually necessary for something. Why should non-mathematicians (or mathematicians) even bother with proofs?? This is quite an interesting discussion.

 

[WindScars]

I haven't much to add anymore, I got my answer. Also my english bad, that makes it difficult to keep a good discussion.

Office_Shredder you didn't get the point too, by the way.

Knowing proofs is not an so-related issue. It is an interesting issue, though, I agree.

I'm definitely changing the paper and ink for text-editor and keyboard on my quest to knowledge.

 

[micromass]

When you are solving an equation or performing a multiplication you are not thinking on your main problem anyway, it actually takes you off the focus. You know it.

True. But you get a feel for the math involved. You get a feel about how hard it is to calculate $\int{x^2dx}$. Furthermore, you can stand in awe of the smart men who made calculating areas so easy!

And I'll bring it up again: what if your calculations involve symbols rather than numbers?? In that case, your computer will be useless. I know computers can do symbolic manipulations but it is still not good enough for if you want to do theoretical physics.

Also (and I'm sure you will disagree with this), there is a certain beauty in the calculations. Actually being able to find the area under a parabola made me almost high when I first done it. After a while it gets boring, but calculations really can be pretty.

 

[micromass]

I'm definitely changing the paper and ink for text-editor and keyboard on my quest to knowledge.

Good luck and let us know how it works out for you! (Seriously, I'm interested)

 

[Matterwave]

At what point do you think "understanding" kicks in? You say "understand what an integral is". Do you simply mean:

I understand the definition of a Riemann Integral:
$$\int_a^b f(x)dx=\lim_{n\rightarrow \infty}\sum_{i=1}^n f(x_i)[x_i-x_{i-1}]$$, and that's it?

 

[WindScars]

Micromass, what do you mean? This is where the computer will exceed. How familiar you are with functional and object-oriented programming? You don't just manipulate symbols, you manipulate objects with a well-defined behavior. Are you working with vectors? Great.

f = function(t) { return t*t; }
vec1 = V3(f(t),(derivative(f))(t),0)
vec2 = vec1.cross(V3_unitx)

Well, you just defined 2 vectors, one whose position is defined by 2 functions of t, one being the derivative of the other, displaying, as result, accelerated and uniform movements for the x and y axis, and other defined by the cross product of the first vector and the unitary x vector. You can visualize it moving on the front of your eyes ready on. There are functions, symbols, objecs. Lots of concepts. No number.

Matterwave, for god's sake, that's the absolute opposite of what I'm saying. Understand it intuitivelly, play with it using programming instead of paper+ink / imagination. Seems like I'm really bad on expressing myself on english (:

 

[bcbwilla]

You're question is rather vague.

You asked: "Are math skills necessary?"

Necessary for what? When you say you are trading in a text editor for a paper and pencil, what is your goal?

 

[WindScars]

bcbwilla read the entire topic. My english is bad but I'm sure this is not vague anymore at this point. Also, I expressed myself horribly, when I used "math skills" to refer to "algebraic skills" from memorizing an integral table to having a collection of tricks to solve equations and all that.

 

[micromass]

Micromass, what do you mean? This is where the computer will exceed. How familiar you are with functional and object-oriented programming? You don't just manipulate symbols, you manipulate objects with a well-defined behavior. Are you working with vectors? Great.

f = function(t) { return t*t; }
vec1 = V3(f(t),(f.derivate())(t),0)
vec2 = vec1.cross(V3_unitx)

Well, you just defined 2 vectors, one whose position is defined by 2 functions of t, one being the derivative of the other, displaying, as result, accelerated and uniform movements for the x and y axis, and other defined by the cross product of the first vector and the unitary x vector. You can visualize it moving on the front of your eyes ready on. There are functions, symbols, objecs. Lots of concepts. No number.

Matterwave, for god's sake, that's the absolute opposite of what I'm saying. Understand it intuitivelly, play with it using programming - as a good old day's mathematician would play on a paper and with imagination (because they had no computer!).

Fair enough. Show me how the computer can be used to verify the substitution law for integrals:

$$\int_a^b f(\varphi(x))\varphi^\prime (x)dx = \int_{\varphi(a)}^{\varphi(b)}f(t)dt$$

I'll be utterly amazed if a computer can do that.

 

[Jorriss]

bcbwilla read the entire topic. My english is bad but I'm sure this is not vague anymore at this point. Also, I expressed myself horribly, when I used "math skills" to refer to "algebraic skills" from memorizing an integral table to having a collection of tricks to solve equations and all that.

Of course you don't want a computer to do all this while you are just learning. You need to build intuition.

 

[bcbwilla]

bcbwilla read the entire topic. My english is bad but I'm sure this is not vague anymore at this point. Also, I expressed myself horribly, when I used "math skills" to refer to "algebraic skills" from memorizing an integral table to having a collection of tricks to solve equations and all that.

I did read the entire topic.

Here is my response:

Sure, you can enter equations into a computer and solve them. But the equations themselves are partially the result of algebraic and other mathematical manipulations. Just look at all of the different formulations of classical mechanics, or quantum mechanics. These equations don't just magically appear in their current form.

 

[WindScars]

Micromass, why would the verification of the substitution law for integrals be useful for an individual pursuing the undestandment of it's universe?

(There are actually many things that could be done for that case, but there's no need to answer this because it's obvious math will always be the best way to solve a mathematical problem. This is not the point.)

I did read the entire topic.

Here is my response:

Sure, you can enter equations into a computer and solve them. But the equations themselves are partially the result of algebraic and other mathematical manipulations. Just look at all of the different formulations of classical mechanics, or quantum mechanics. These equations don't just magically appear in their current form.

These equations are not quantum mechanics. These equations are an algebraic model of quantum mechanics. It can be modeled by matters of objects and functions aswell.

 

[micromass]

Micromass, why would the verification of the substitution law for integrals be useful for an individual pursuing the undestandment of it's universe?

...


Excuse me, but what physics classes did you already take?

 

[WindScars]

Just answer.

 

[bcbwilla]

These equations are not quantum mechanics. These equations are an algebraic model of quantum mechanics. It can be modeled by matters of objects and functions aswell.

Objects and functions representing what? The equations!

 

[Jorriss]

Just answer.

I'm curious to now. What actual background do you have?

Mathematical details are enormously important to physics. Operators not commuting is at the heart of quantum mechanics. Schwarz inequality leads to the uncertainty relationship. And those are just the most basic details of quantum. Mathematical details are hugely important to physics.

 

[WindScars]

You are asking this because you are suspecting I am not good at physics. You want to be sure I know what I'm talking or else it won't make sense arguing. This shows that, for you, who I am is relevant to this discussion. Unfortunatelly, whoever my human incarnation is completely irrelevant to any of the laws of physics. It's funny how you know how to do an integral but don't understand this basic concept.

Consider myself as a 10 yo who never did any math. You'll still have to direct to my arguments, though.

Mathematical details are enormously important to physics. Operators not commuting is at the heart of quantum mechanics. Schwarz inequality leads to the uncertainty relationship. And those are just the most basic details of quantum. Mathematical details are hugely important to physics.

This is certainly fundamental to the human model of quantum mechanics. No doubts of that. Edit: this still has nothing to do with the topic.

 

[micromass]

Just answer.

If we want to solve the Schrodinger equation, then we might want to apply separation of the variables. This will yield a differential equation that will be solvable by integration. This last step involves solving a separable differential equation. That separable equations can easily be solved is a direct consequence of the substitution law.

Just one of many examples.

 

[WindScars]

And why would you need a paper and algebraic skills to do this?

 

[micromass]

And why would you need a paper and algebraic skills to do this?

Because a computer can't?? I asked you before if a computer could verify the substitution law. You didn't respond to that.

 

[Jorriss]

And why would you need a paper and algebraic skills to do this?

Because computers don't know how to solve differential equations numerically (the ones they can even solve numerically without a person using some algebraic input) other than immensely simple integrals.

Do you know that the best way to solve the quantum harmonic oscillator is by strictly algebraic means?

You are asking this because you are suspecting I am not good at physics. You want to be sure I know what I'm talking or else it won't make sense arguing. This shows that, for you, who I am is relevant to this discussion. Unfortunatelly, whoever my human incarnation is completely irrelevant to any of the laws of physics. It's funny how you know how to do an integral but don't understand this basic concept.
.

...Jesus...

People ask because you give the impression of being clueless about physics and math and don't want to talk to a brick wall.

 

[WindScars]

Jorriss, yes, it can.
About your edit, so don't lose your damn time. You'll not get anything from me and you can consider my background that of a 10 years old if you want to.

Micromass, a computer can verify and aply the substitution law. It can find it out, too, but this is harder.

By the way, what will solving the schrodinger equation do to us? I'm asking it, yes, I have no idea.