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Maths and the real world.

  
 
Wombat
18:30 / 21.10.03
Is there a deep connection between mathematics and the real world?
Some of mathematics predicts things long before any experimental results have proved the case. ( both QM and GR to take recent examples)
Some is totally useless. (hell most)
You can prove with mathematics that mathematics can`t cover everything. (godels theorem)
BUT can a mathematical model predict the real world close enough in ALL cases to provide a useful tool? Are there situations where it should be dropped and a better tool used in it`s place. (Wolfram)
Some scientists say they have a deep intuition for what is true and what is not. (hawkings `know the mind of god`, penrose and plato)
Do we only see descriptions of the universe in terms of numbers because of their usefulnes so far?
Are we blinded by mathematics?
OR is there some deep underlying connection between numbers an the real world?

My current viewpoint is that mathematics is a descriptive view.
There are enough possible viewpoints encompassed in number that the real world must be there somewhere. Geniuses simply describe their worldview to the masses in terms of mathematics.
It`s simply a very good descriptive view. People understand the real world and then describe it.
Although the usefulness of mathematics in prediction does give me pause for thought.
 
 
SMS
21:09 / 21.10.03
One possibility is that our mind contributes something to the structure of the universe. The theory goes something like this.

To our minds is presented the manifold of experience (the stuff that we don't contribute to experience...a jumble of unordered stuff...what an oyster might see)

Our mind then orders this jumble in space and time, also unifying it all into ONE consciousness (namely mine)

The actual experiences we have are thus, of there nature, geometric, and hence, mathematical.
 
 
Lurid Archive
22:19 / 21.10.03
BUT can a mathematical model predict the real world close enough in ALL cases to provide a useful tool? Are there situations where it should be dropped and a better tool used in it`s place. (Wolfram)

Wolfram did say that, didn't he? Thing is, its exaggeration. Wolfram decided that math that is *traditionally* used in sciences was inappropriate and wanted instead to use math that is less traditional.

This sounds like a pedantic point, but it really isn't. Math is not limited to bank interest and so forth. It is the systematic and abstract study of structure and symmetry as well. So any time you want to study the world in some detailed and systematic way you are probably using or developing math. Its important to remember that math is not set in stone, but is rather a certain approach to problem solving.

The large exception to the utlity of math seems to be in any problem involving human affairs, essentially because everything involving people is too difficult for that kind of approach. Physics is, in a certain sense, much easier than politics, say. Which is why math is very useful for one and of quite limited use for the other.
 
 
w1rebaby
00:01 / 22.10.03
I think you need to think this one out a little more.

"mathematics is a descriptive view" - as opposed to what, a prescriptive view? Mathematicians telling the world it should behave in a certain way? Of course it's descriptive. The issue is, is it capable of adequately describing the world, and to what degree?
 
 
SMS
02:49 / 22.10.03
And is math discovered or invented?
 
 
Perfect Tommy
04:38 / 22.10.03
Math is the direct study of patterns, without there needing to be 'stuff' attached.

Nature exploits whatever patterns it can. So increasing our understanding of patterns is often going to predict natural law.

As to whether math is invented or discovered... *shrug*. Mathematical entities with no physical analog get invented all the time, but often a corresponding pattern in the real world will be discovered later. I think imaginary numbers are one example--it doesn't make any 'real' sense for the square root of -1 to exist, but imaginary numbers are used all the time in very real electrical circuits.

I guess the answer to the question, "Is math discovered or invented?" is 'yes'.
 
 
cusm
14:59 / 22.10.03
More directly, Math is a language. But moreso, a language undependent of human understanding, universal in application and appeal. It offers a meta-language by which all things can be described, for all things in nature can be expressed in numbers of some form. Its in the interpretation of those numbers and relations that individual understanding can differ. But as far as one can go to use of a language of "truth", math is as close as we can get.
 
 
Scrubb is on a downward spiral
16:58 / 22.10.03
Random thoughts expunged: there are aspects of maths which occur and can be applied to the real world - the golden ratio, prime numbers (13 will be a prime whichever side of the galaxy it's on...).

And there are areas of maths contained in defined boundaries - Euclidean geometry fr'example - which can be manipulated to reveal the truth within those boundaries...

And there are the fights and shifts when the above boundaries are defined. Thinking in particular of negative and imaginary numbers, and their acceptance into the mathematical community. And thinking of the young man in Pythagorus' school who was murdered for suggesting that negative numbers exist.
 
 
Wombat
22:28 / 23.10.03
SMatthewStolte- Mathematics can contribute to my perception of the universe. The idea of mathematics changes the universe. My brain/mind/whatever colours my perception...this has practically no effect on the universe.I agree that our experience is `of nature`. BUT I don`t unerstand how this is `of geometry`. geometry (in my current worldview) is created by people. But people who assume it is fundamenatal to the way the universe works seem to get damn good results.

Lurid- Fair play. Wolfram sold out on his own concept. BUT if anything new came along that wasn`t covered by mathematics then I suspect that six months later there woul be a new form of mathematics that encompassed it. Mathematics eats it own children to grow. Are there any concepts that can`t be covered by mathematics?

Fridgemagnet- MY current view is that mathematics is decriptive. Other views I`ve stumbled across are-
a) the universe is pure mathematics. our percieved universe is it`s shadow.
b) the architect of the universe used maths as a design.
c) the prescriptive view you described.
(I`m not gonna defend any of these views..they give me giggly fits.)

and many more.
I believe that mathematics is a description.
BUT it`s only a belief. I might think it`s descriptive. Lots of people I respect disagree.
I know I need to think it out more. That`s why I`m requesting help.

Tommy, Cusm, Scrubb - A lot of math is invented all the time. Some of it is usefull in real world situations. We pick and choose the maths that is useful in our place, time and head/culture space.
As for undependant of human understanding- agreed. There is a lot of stuff I can do the mathematics for. That provides real world results. I don`t pretend to understand it.

Hence my question - I can solve real world problems by manipulating symbols with no real understanding. Follow the rules - they always work. (?)
It`s possible that someone can calculate the progression of an electron thru an energy barrier to a very small degree. But be unable to define electron or energy barrier. Practical result is the transistor.

People predict impossible things with maths.

some ex patent clerk thought that for something to be true it had to be true all the time. Used a bit of topology math that regarded both space and time as mallable.
At the time space and time were separate entities.
result of this was a bending of light near large masses.
20 years later bending was observed.

Some nutter noticed that the energy transitions in atoms are much ` nicer` if they happen in small digital quantities.
result of this is the transistor.

At the moment some bloke has simplified the equations of the stanard theory by assuming extra rolled up dimensions. (11 dimesional universe..3 space..one time..rest un-observable))
It looks nice. It`s easier than the current model. BUT going by current experience maths that looks nice and is simple turns out to be the most usefull in the real world.

I don`t have a problem with easy mahts being a usefull model.
I can`t see any way that humans have evolved to find usefull mathematic nice.

'nice'...looking at stars is nice...watching wind on forest is nice..mathematics is EXACTLLY the same sort of nice..I don`t know why. I wanna know why..I suspect learnt happy is the same sort of happy. A conformity rush.
 
 
SMS
00:56 / 24.10.03
Some people think that belief is not in the intellect, but rather a feeling that we have. On top of this, we believe in deductive reasoning (when we do it) because of an ease of transition from one step in the deduction to the next. That ease of transition of thought produces a kind of pleasure or niceness to the conclusion of the deductive conclusion. Thus, those theories that seem to us to be MOST TRUE are thus the ones by which we can make the easiest transition of thought (or derive the most sense of niceness from). I don't know if I believe this, but I know that one of those famous early modern philosophers did. I'll think of his name soon.
 
 
Quantum
10:21 / 24.10.03
Are there any concepts that can`t be covered by mathematics?
Erm, a bazillion? Human interaction, dreams, anything with an adjective in? The Humanities, history, comedy etc. etc. etc.

I think Mathematics is a human construct based on our perception of identity- we see an apple, we say One apple; cut it in half, is it two things or two halves? In the world, there are no numbers.

As math is described as a language, that implies it's made by people- I don't hold that it is the language of 'truth', despite it's consistency and reliability, otherwise it wouldn't need to be learned. The notion of synthetic A Priori knowledge (eternal truth we have to learn) is dodgy IMO.
 
 
SMS
14:06 / 24.10.03
That isn't the way I understood synthetic a priori truth. I thought it was something like this

2+3 = 5

is not analytic because, in the human mind, when you are thinking of five, you aren't thinking of the same thing as 2+3. The difference is subtle but there. Consider (2395-2390). Someone here wasn't thinking of 2395-2390 when they read 2+3, but they are equal. Using set theory, of course, you can define 2+3 as a set containing exactly the same elements as the set 5, and, if you really think of the operation in this way, you can probably get your mind to think of 2+3=5 in an analytic way.

All Bachelors are Unmarried Males

Is analytic a priori because when you think of a bachelor, you are thinking of exactly the same thing as an unmarried male. The concept of unmarried and the concept of male are contained in the concept of a bachelor.

Both of these facts are both learned and a priori. Children aren't born with this knowledge about the word bachelor or the concept of a bachelor, but it is a priori truth because it is impossible to think otherwise. If you get the concepts in your head clearly, you can't think that bachelors are married males. If you are thinking something LIKE that, you aren't thinking of bachelors. You're just using the word bachelor to describe something else (or using the word married to describe something else). Same thing goes for 2+3=5. If you think that 2+3=4, you aren't really thinking of 2+3 or you aren't thinking of 4 or you aren't thinking of "+" correctly.

On the other hand, you can grasp the concept of bachelor perfectly well and think they all drive pickup trucks and be right or wrong about that. That's synthetic a posteriori.
 
 
angelvanilla
06:50 / 27.10.03
"All Bachelors are Unmarried Males

Is analytic a priori because when you think of a bachelor, you are thinking of exactly the same thing as an unmarried male. The concept of unmarried and the concept of male are contained in the concept of a bachelor."

Not really if I am thinking of bachelors as possible future mates of some group of single women and that's the only way I think about it. Then I don't need to think about "unmarried males" at all to think about bachelors.
 
 
Quantum
09:15 / 27.10.03
I think it's dodgy because of the a priori/a posteriori, analytic/synthetic, logical/empirical, necessary/contingent distinction. If you can have synthetic a priori truths then why not analytic a posteriori truths, empirical facts gleaned from tautologies and the like?

I think there are a priori/analytic/logical/necessary truths that concern logical constructs like mathematics and language, and often tautologies (always?), contrasted with a posteriori/synthetic/empirical/contingent truths learned about the world, and never the twain shall meet. The first category (including math) tells you nothing about the world of objects, just about the rules we apply to the world.
 
 
Lurid Archive
09:57 / 27.10.03
Are there any concepts that can`t be covered by mathematics?

I agree with Quantum on this one. Most concepts that a person will have are nothing to do with math.

Interesting last post, Quantum. Not sure I have either a strong nor informed opinion about it, really. But I think that your statement

The first category (including math) tells you nothing about the world of objects, just about the rules we apply to the world.

is partly the point of the question, surely? You use maths to model a physical situation, say, and arrive at some non-trivial yet tautologous conclusion that lets you construct a bridge. As a for instance. Does the fact that the bridge doesn't fall down tell you something? The point being that the logical and empirical have interacted.

One last point is to do with math as a language. I'm not saying that I disagree with the proposition, but I don't quite see it as self evident either. True, lots of math is symbolic, but then lots isn't. As noted above, most concepts cannot be stated in "math-speak" and besides much of "math speak" is in english.

I guess my question would be how to decide when a specialised vocabulary and shorthand become a "language".
 
 
SMS
14:46 / 27.10.03
Not really if I am thinking of bachelors as possible future mates of some group of single women

If you are thinking about bachelors in this way, then you are thinking of something other than the bachelors I was talking about. You just assign a different meaning to the word bachelor.
 
 
Quantum
15:21 / 27.10.03
Does the fact that the bridge doesn't fall down tell you something? The point being that the logical and empirical have interacted. Lurid

The interaction I don't deny, you can put empirical data into a math model and get other empirical data out; but pure math can't tell you new things about the world, only new things about math.
In real life of course we use both modes of thought as and when we want- for ease of reference I'll call the first sort Deductive and the second Inductive. We translate information from one mode to the other but they are different;
We measure the girders and the chasm inductively by going there with a measure, changing the distances into numbers then we apply the math model (which is independent of instance, so we can sit at our desk and do it or our descendants can do it on the moon and get the same result) and deduce some other numbers, which we translate back into measurements and build the bridge- at no point is the chasm a number, or the math in the world (obviously).
Hmm. I'm not expressing myself very clearly, let me think about how to explain what I think and get back..

SMattStolte is using Bertrand Russel's interpretation (if memory serves) but my objection is that analytic and a priori knowledge are identical and synthetic and a posteriori knowledge are identical- so synthetic a priori is equivalent to a priori a posteriori, which makes no sense.
I'm willing to be convinced otherwise.
 
 
Quantum
15:26 / 27.10.03
when you are thinking of five, you aren't thinking of the same thing as 2+3
But if we were cleverer or thought harder about it we could. What I mean is even though we can't immediately grasp whether or not 5467389 is a prime doesn't mean it's synthetic, it's still analytic because you don't need any extra information to find out. You can analyse it and get the truth from it alone (maybe with a computer) whereas synthetic truth you can only discover by adding other truths to it.

angelvanilla, instead of the bachelor example consider "a triangle has three sides" or "red is a colour", same argument.
 
 
angelvanilla
18:12 / 27.10.03
“I think it's dodgy because of the a priori/a posteriori, analytic/synthetic, logical/empirical, necessary/contingent distinction.”

Many of these distinctions have been discredited on contemporary views. It is hard for me to imagine how there is any knowledge without experience.

“Most concepts that a person will have are nothing to do with math.”

That seems right. But I feel all of those exist as shells over maths. Maths are the language of bodies in motion or at rest.

It seems that some people are forgetting that math can only exist if there is some system of objects. Math doesn’t come from nowhere. It comes from the way we interact with other objects.

“If you are thinking about bachelors in this way, then you are thinking of something other than the bachelors I was talking about. You just assign a different meaning to the word bachelor.”

Yes, and then your “analytic truth” is no longer that way to me; thus, it is not really an analytic truth. It becomes the same sort of truth as your number example. Someone might define 5 as 2 + 3, and another as 123 – 118. In the math case both are necessary inside the system of numbers, and in the English case, both definitions of “bachelor” are necessary inside the system of English.

“angelvanilla, instead of the bachelor example consider "a triangle has three sides" or "red is a colour", same argument.”

The first is a necessary mathematical truth & the second a necessary truth of English.
 
 
SMS
20:26 / 27.10.03
Someone might define 5 as 2 + 3, and another as 123 – 118.

I'm not saying that you can't think of 5 in a way that every time you are thinking of 5, you are thinking of 2+3. It's just that you don't actually do it, most of the time. I imagine you can think of a bachelor in some way such that B=UM is also synthetic a priori, but I'd have to think about how you could do that. Something along the lines of prospective mate for an arbitrary, single, heterosexual female might be a start.

Many of these distinctions have been discredited on contemporary views.

More evidence in my mind that these contemporary views are false. If I recall, one of these arguments makes a claim that you can't make sense of the concept of a "meaning," because you can't reduce it to experience.
 
 
angelvanilla
06:56 / 28.10.03
"I'm not saying that you can't think of 5 in a way that every time you are thinking of 5, you are thinking of 2+3. It's just that you don't actually do it, most of the time."

Oh I don’t know. I probably think of 5 as 3 and two together more often than any other way. That’s likely a trace to the way many of us learned to count and do basic addition. Two apples and three apples is five apples.

"I imagine you can think of a bachelor in some way such that B=UM is also synthetic a priori, but I'd have to think about how you could do that. Something along the lines of prospective mate for an arbitrary, single, heterosexual female might be a start."

I guess I am saying, if you want to stick to those sorts of categories, that to me it seems as if all knowledge is synthetic and that there is no a priori or a posteriori when it comes to knowing. That division is a matter of our linguistic systems and not anything about knowledge.

"If I recall, one of these arguments makes a claim that you can't make sense of the concept of a "meaning," because you can't reduce it to experience."

Maybe, I don’t know. Sounds obvious and at the same time silly if you ask me! Just because something can not be reduced down to experience, maybe you’ll agree, does not me it has no sense at all! Ideal equilateral triangles don’t occur in experience, and yet, they make perfect sense! I have many reasons why I think those divisions are no good. Some of them concern the fact that some male figured "bachelors are unmarried males" is a paradigmatic example of a priori knowledge while women were being treated like objects.

I like the contemporary picture a little more than I like the archaic one.
 
 
Quantum
10:16 / 28.10.03
all knowledge is synthetic and that there is no a priori or a posteriori when it comes to knowing.
Fair position, I agree with the first part- do you accept 'synthetic' is synonymous with contingent, empirical and a posteriori? When you say there is no a priori or a posteriori, don't you mean there's only a posteriori, knowledge learned in life?
And do you see any useful distinction between necessary mathematical truths and necessary linguistic truths? Is there a difference between "2+2=4" and "Two plus two equals four"?
 
  
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