Exploring Math's Structural Impact on Logic & Beyond

[wildmandrake]

Is the struggle to reach beyond relativity and quantum mechanics, beyond the big bang, possibly its creation and necessity to our logic, the result of structural problems with mathematics itself? The fundamental unit of maths is numbers, is it not? The discovery of zero or its inclusion in the number system (which ever way you like call it) as an origin point for starting to count is necessary for making sense of our whole system. But it leads to some natural consequences for the development of logic and therefore maths. It seems to me that all kinds of logical tricks are being used to get around this basic truth - multiverses, string theory, etc Maybe we need to consider that each move to a new understanding, each paradigm shift that gave us a new view meant a widening of the number system, or definition of what is actually being counted or how it is counted/measured. the problem is we take our eyes with us where ever we go and maths is our eyes, is it? Affecting the way we look at things and what we see as a reasonable problem and answer?

 

[Crosson]

The actual point is quite vague, "structural problems with mathematics itself". It seems as though you are talking about mathematics in general without really thinking about what it is, and what sort of "structure" is involved.

The only remotely concrete example you give is zero; to suggest that zero poses a structural problem suggests a complete ignorance of modern algebra. Please give more specific examples.

 

[Chronos]

That is a nice, pseudo logical argument, but it has no scientific content. As Crosson suggested, you need to make a specific point.

 

[neurocomp2003]

"structural problems with mathematics itself". its not math its the human perceptoin of math that is flawed =]

 

[arildno]

There are no limits on maths, only on the creativity and ability of mathematicians.

 

[Alkatran]

There are no limits on maths, only on the creativity and ability of mathematicians.

Math can't be not math. If we can prove that nothing is not math, we have a proof by contradiction that math is everything.

 

[wildmandrake]

It is interesting that mathematicians are so evangelical about the universality of math, it is a symbolic language of description like any language. i admit is is special because it uses physical systems as its basis or at least the numerical relationships, but numbers and logic are limited and those limitations are important to the kind of things that system can handle. I am ignorant of modern algebra I'm just an average joe interested in understanding how the world works and I'm effected by the things people using math come up with. I'm not ignorant but my expertise, if i have any, is in the realm people not the technical sciences or theory associated with them. If mathematicians aren't looking at its limitations then they will be limited by them. As for other examples well I'll get back to you. I'm specifically interested in the possible connection between the big bang theory and math, because it seems to be where maths, physical laws break down.

 

[arildno]

Math can't be not math. If we can prove that nothing is not math, we have a proof by contradiction that math is everything.

 

[arildno]

wildmandrake:
Well, since you admit you're ignorant of maths, you're really not the one qualified to draw the boundary between what math can do, and what it can't do.

 

[maverickmathematics]

Or are we talking about the axioms Mathematics is based on (typically Peano's)? And the limits this places on mathematics?

-M

 

[arildno]

Or are we talking about the axioms Mathematics is based on (typically Peano's)? And the limits this places on mathematics?

-M

Minor comment:
"Or are we talking about the axioms Mathematics CAN BE based on (typically Peano's)?"

 

[wolram]

Is the struggle to reach beyond relativity and quantum mechanics, beyond the big bang, possibly its creation and necessity to our logic, the result of structural problems with mathematics itself? The fundamental unit of maths is numbers, is it not? The discovery of zero or its inclusion in the number system (which ever way you like call it) as an origin point for starting to count is necessary for making sense of our whole system. But it leads to some natural consequences for the development of logic and therefore maths. It seems to me that all kinds of logical tricks are being used to get around this basic truth - multiverses, string theory, etc Maybe we need to consider that each move to a new understanding, each paradigm shift that gave us a new view meant a widening of the number system, or definition of what is actually being counted or how it is counted/measured. the problem is we take our eyes with us where ever we go and maths is our eyes, is it? Affecting the way we look at things and what we see as a reasonable problem and answer?

I think he is saying something like---
Maths can describe black hole evaporation via Hawking radiation, and yet
we have little possibility of finding out if it correct, it may be a model with
no reality.
Or math can describe quantasized space time, "or partly", and again it may
be wrong.
It is not the math that is wrong, it is how it is used, and testing any
prediction it makes.

 

[ohwilleke]

Its worth noting that to some extent modern physics relies on math that does restructure traditional mathematical assumptions. For example, an important basis for much of modern physics (by which I mean post-classical physics, not simply physics conducted recently) is abstract algebra. Abstract algebra, such as group theory and non-communitive algebras, basically involves taking the rules of algebra that you learned in high school and changing them. For example, in some varieties of group theory, it is possible to choose an A and B such that A+B=A. And, in non-communinative algebras A*B will not necessarily equal B*A. It turns out the the algebras have real physical applications.

General relativity is based on non-Euclidian geometry, which has different assumptions from those found in ordinary high school geometry.

Likewise, another major deviation from ordinary high school mathematics, which is complex analysis (i.e. calculus involving the possibility that numbers can have imaginary components, a.k.a. complex numbers) also has physical usefulness.

Another common field of exploration in modern physics theory, at least, is the possibility that space and time are not continuous, which is an assumption of all mathematics conducted on the set of real numbers or the set of complex numbers (or subsets of them) as mathematics generally is now.

In short, mathematics has shown no inability to measure up to new possibilities in fundamental and structural ways to address new problems in physics. This isn't to say that whole new classes of mathematical structures might be developed some day. A fairly recent example is the resurrection of the notion of a non-integer dimension, used in fractal and chaos mathematics, which had lied dormant for many decades before being rediscovered in the 1980s. Other new ideas could arise.

But, I strongly doubt that any roadblock is beyond mathematical expression entirely, and while our ability to calculate the results implied by our theories at time actually falls behind our ability to do experiments (for example in parts of QCD), I don't think that we are at such a roadblock now.

 

[arildno]

Good post, ohwilleke.
Personally, since I think we might replace every instance of the word "maths" with the phrase "good and productive thinking", I have rather negative view on phrases like "limits on math"..

 

[wildmandrake]

ohwilleke thank you for an intelligent and unemotional response, same to the others on this list. For those of us who have limited maths and a real curiosity but are still capble of "good and productive thinking" (thanks for that line arildno) without it. As a poet I'm aware of a variety of ways of thinking/feeling that seem in my limited knowledge not easy to fit into math and yet are still important and productive for insight. I look at some of the theories as mentioned about the hawking radiation and other things, like the big bang, string theory and the brane with multiverses and wonder about them when there seem to be simpler solutions possible like an eternal universe. Now I have heard about the acceleration of the universe and people madly looking for some explanation but some this stuff seem as sloppy as New Age thinking

 

[z0r]

Mathematics as an expression of a culture/civilization

I found this post interesting because at the moment I am reading Spengler's Decline of the West. I have just finished the introduction and started the second chapter: Meaning of Numbers.

Spengler argues that the idea of a universality of mathematics is nonsense; that number is one thing, but the methods developed in mathematics are not developed unless they are expressions of "the soul of a culture," so to speak.

He compares the Romans and Greeks, which he refers to as the Classical culture, to European culture, which he refers to as Western culture. He claims the men of the Classical culture had expressed their mathematical ideas long before Pythagoras in their architecture, which was rigid and of substance. The mathematics of the Classical civilization, he argues, was one of solid substance. The mathematics of the West is one of space.

Spengler mentions Euclid's definition of a line as "length without breadth." Spengler claims this is a "pitiful" definition to Western civilization, but fit perfectly well into the Classical worldview.

I have found The Decline of the West to be a fascinating critique of many things I had once considered "obvious" or taken for granted. My initial impression is one of majestic awe... this is indeed a work not to be taken lightly.

Has anyone else read/heard of it?

 

[arildno]

The major work of a Nazi crackpot like Spengler?
Sure I've heard about it..

 

[selfAdjoint]

The major work of a Nazi crackpot like Spengler?
Sure I've heard about it..

Spengler was neither a Nazi nor a crackpot. He published the first edition of his study of history, The Decline of the West before WWI, when Hitler was still trying to become a painter, and his conclusions in it contradict the Thousand Year Reich, as you can see from the title.

 

[arildno]

1)Sure he was a crackpot; "organic" theories of societal evolution like Spengler's, Toynbee's and Hegel's (and, for that, matter, Marx) have been thouroughly debunked; history doesn't allow for such over-simplified schemes of explanation.

2) He certainly was a racist and anti-democrat, although he disdained the "vulgar" appeal of the full-blown Nazis.
He was, however, not entirely unsympathetic to them.

 

[juvenal]

ohwilleke thank you for an intelligent and unemotional response, same to the others on this list. For those of us who have limited maths and a real curiosity but are still capble of "good and productive thinking" (thanks for that line arildno) without it. As a poet I'm aware of a variety of ways of thinking/feeling that seem in my limited knowledge not easy to fit into math and yet are still important and productive for insight. I look at some of the theories as mentioned about the hawking radiation and other things, like the big bang, string theory and the brane with multiverses and wonder about them when there seem to be simpler solutions possible like an eternal universe. Now I have heard about the acceleration of the universe and people madly looking for some explanation but some this stuff seem as sloppy as New Age thinking

You cannot understand a physical theory in a meaningful fashion if you do not understand the mathematics. The job of a theoretical physicist is to quantify physical phenomena.

And discussing or speculating about a theory of which you have a superficial understanding just typically leads to nonsense, as you would naturally expect.

 

[ohwilleke]

ohwilleke thank you for an intelligent and unemotional response, same to the others on this list. For those of us who have limited maths and a real curiosity but are still capble of "good and productive thinking" (thanks for that line arildno) without it. As a poet I'm aware of a variety of ways of thinking/feeling that seem in my limited knowledge not easy to fit into math and yet are still important and productive for insight. I look at some of the theories as mentioned about the hawking radiation and other things, like the big bang, string theory and the brane with multiverses and wonder about them when there seem to be simpler solutions possible like an eternal universe. Now I have heard about the acceleration of the universe and people madly looking for some explanation but some this stuff seem as sloppy as New Age thinking

I would distinguish speculative and sloppy.

The bottom line issue is that simple solutions do not work. Life would be great if you could explain all there was to know about gravity with F=GMm/R^2. Alas, nature has not been cooperative. It turns out that to correctly determine the effects of gravity you need godawful rank two tensors, pressures, and a rethinking of the basic nature of time and space and all other sorts of stuff that is hard to get a handle on even with a graduate education in physics.

The famous line "who ordered that?", pretty much sums up quantum mechanics. Back when we had just protons, neutrons, electrons and classical electromagnetic fields we were pretty proud of ourselves. This plus gravity explained virtually everything our eyes could behold. At the tail end of the 19th century the were talking about all future advances in physics coming in the nth digit.

Then, a few slight cracks appear in the edifice. In a single lifetime, we go from the simple models of classical physics to a world with six kinds of quarks, three colors, three kinds of neutrios, two additional fundamental forces, three kinds of electrons, gluons, Ws, Zs, photons and probably a few malingerers that we haven't found yet. And, don't get me started about what a revolutionary concept it is to go from the deterministic, continuous classical physics universe to a universe where absolutely fricking everything in the universe is random in an extremely pure sense and physical phenomena act differently when people are or are not looking at them.

And, neither Hubble nor anyone else expected to discover the Big Bang. The whole eternal universe gig was looking pretty good in 1900. Keep in mind that the very term "Big Bang" was invented as a term of derision by a Big Bang skeptic promoting a steady state model of the universe, and not by Big Bang proponents.

When experience repeatedly shows you that the universe is far more weird than you'd ever expected it to be, naturally, you become more open to weird approaches to explaining what is out there.

Are people madly looking for solutions? Yes.
Will the theories on the cutting room floor when we're over look like the remains of a ticker tape parade? Yes they will.
But, every one of these published theories is a plausible, fairly rigorous, connected to some heartland of physics approach.

OK, some of the theories are a little over the top. No, lots of the theories posit really strange things. And, as a general rule one wants to avoid believing in theories that involve really strange things (not that the humanities have ever had a problem with really strange things). But, I sincerely believe that somebody is on the right track.
If you want to know which theories are most on the right track, contact your local bookie.

 

[juvenal]

Are people madly looking for solutions? Yes.
Will the theories on the cutting room floor when we're over look like the remains of a ticker tape parade? Yes they will.
But, every one of these published theories is a plausible, fairly rigorous, connected to some heartland of physics approach.

OK, some of the theories are a little over the top. No, lots of the theories posit really strange things. And, as a general rule one wants to avoid believing in theories that involve really strange things (not that the humanities have ever had a problem with really strange things). But, I sincerely believe that somebody is on the right track.
If you want to know which theories are most on the right track, contact your local bookie.

That's the thing. Crackpots complain about how mainstream physicists shut them out, as though there were some deliberate plot to suppress new theories. In reality, however, there are plenty of new, sometimes controversial theories put out by mainstream physicists. These theories are usually based on legitimate physics, and are just awaiting experimental verification.

 

[Thor]

There are no limits on maths, only on the creativity and ability of mathematicians.

The nature of math is self limiting by definition. It is quantitative in nature which means it depends on limits and definition. Math is somewhat useful in dealving into the quantitative aspects which pertain to qualitative study, but qualitative analysis is beyond the realm of math. And it utterly fails as a tool in the exploration of the undefined - i.e. infinity.

 

[Alkatran]

The nature of math is self limiting by definition. It is quantitative in nature which means it depends on limits and definition. Math is somewhat useful in dealving into the quantitative aspects which pertain to qualitative study, but qualitative analysis is beyond the realm of math. And it utterly fails as a tool in the exploration of the undefined - i.e. infinity.

ummm have you ever done Math? Infinity is a MATHEMATICAL concept. Undefined is also a mathematical concept. Math is about analysing logical structures. If you can't do something because of the definitions, change the definitions (example: abstract algebra, apparently).

And all "qualitative" things are quantitative. Just because you don't know the exact numerical value doesn't mean there isn't one.

 

[Mike2]

The nature of math is self limiting by definition. It is quantitative in nature which means it depends on limits and definition. Math is somewhat useful in dealving into the quantitative aspects which pertain to qualitative study, but qualitative analysis is beyond the realm of math. And it utterly fails as a tool in the exploration of the undefined - i.e. infinity.

The thing missing in mathematics is HOW to use it. As it stands, we humans have to think intuitively on just what it is we are trying to prove with math. Reality and the laws of physics don't (at this time) naturally fall out of the math. In fact we would not even know where to begin. So there is something apart from mathematics needed to discern reality. And that is where math is limited.

 

[Hurkyl]

qualitative analysis is beyond the realm of math

I'll have to echo Alkatran -- have you ever done math?

 

[Chronos]

The trouble with mathematical solutions is they can give unphysical results. Not every possible solution has an equivalent in the real world [e.g., string theory]. This is a huge problem, and why we have particle accelerators and telescopes.

 

[Thor]

I'll have to echo Alkatran -- have you ever done math?

Yes - BS (of course) with minor in physics - 3.6GPA (out of 4). Of course that was 1972 and today I'd have trouble distinguishing a LaPlace transform from a Fourier series.

There is a very real difference between quantitative analysis and qualitative analysis. There are areas of both which are interrelated, but discount the distinction and try to solve EVERYTHING quantitatively and you may as well try to hammer a nail with a toothbrush.

The three criteria used in logical examination are quantity, quality and dimension (relative position and configuration). Don't they teach this today?

Mathematics - our most exact science - deals mostly with quantitative values. But equations also have QUALitative properties.

Consider the simple equation: (+1)+(-1)=Ø

The quantitative value 'Ø' is absolute - neither positive or negative. But any other value such as |1| must be assigned a positive or negative quality in order to have meaning to the function. If you don't believe trying to get a quality to perform like a quantity can be a problem, consider the equation :

sqrt -1

 

[Thor]

ummm have you ever done Math? Infinity is a MATHEMATICAL concept. Undefined is also a mathematical concept. Math is about analysing logical structures. If you can't do something because of the definitions, change the definitions (example: abstract algebra, apparently).

Yes, mathematics trys to deal with infinity. It does so poorly. Logic REQUIRES definition and infinity is UNdefined.

 

[turbo]

If you don't believe trying to get a quality to perform like a quantity can be a problem, consider the equation :

sqrt -1

Actually, mathemeticians have found interesting uses for imaginary numbers (square root of a negative number). Consider the Mandelbrot set, which is defined in both real and imaginary-number coordinates.

 

[arildno]

If you don't believe trying to get a quality to perform like a quantity can be a problem, consider the equation :

sqrt -1

It seems you have difficulties on a much more fundamental level than being unable to distinguish between a Laplace transform and Fourier series.

 

[Hurkyl]

Yes - BS (of course) with minor in physics - 3.6GPA (out of 4).

Then I'm greatly surprised that you consider qualitative analysis outside the realm of mathematics. Here's a short list of qualitative concepts that are firmly entrenched in mathematics, and I'm only drawing from analysis: continuity, compactness, connectedness, orientation, denseness, sparseness.

Incidentally, how would you classify doing arithmetic in the complexes, quaternions? Or worse, arithmetic in an abstract group or ring? Or arithmetic on ideals? What about doing anything in graph theory?

The quantitative value 'Ø' is absolute - neither positive or negative. But any other value such as |1| must be assigned a positive or negative quality in order to have meaning to the function.

This is curious, because I would say that -1 and +1 are both values, (in particular, they're both real numbers) and I'm sure anyone with whom I work would agree.

If you don't believe trying to get a quality to perform like a quantity can be a problem, consider the equation :

sqrt -1

And this just boggles me for several reasons:

(1) It's not an equation.
(2) How is it trying to get a quality to perform like a quantity?
(3) Why do you think this expression is a problem?

 

[Crosson]

Being the square root of negative one is a quality, and Thor was expressing the view that the imaginary unit is not a quantity.

Mathematics is not qualitative, but humans doing mathematics use qualitative thinking.

 

[arildno]

Being the square root of 53 is also a quality.

 

[Alkatran]

Being the square root of 53 is also a quality.

Aren't qualities just 'good' properties?