Exploring Math's Structural Impact on Logic & Beyond

[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?

 

[Crosson]

Qualities are more general then quantities. In fact, quantities have qualities.

When someone says that mathematics is not qualitative, they mean that it does not make statements (qualitative or otherwise) about non-quantities.

 

[Hurkyl]

And they'd be wrong, because mathematics says lots about things that aren't quantities.

E.G. topological spaces, groups, rings, categories, graphs, functions, sets...

and those are just "nouns"! Math has lots to say about the "adjectives" one can apply to "nouns" too, such as connected, abelian, continuous, cartesian, cyclic, etc.

 

[Thor]

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?




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.

|1| is a quantitative property common to both +1 and -1. Only the property of positivity or negativity qualifies those values as NOT being the same.

And this just boggles me for several reasons:

(1) It's not an equation.

Sorry haste makes for misunderstanding (and one-sided equations which are illogical) -

(2) How is it trying to get a quality to perform like a quantity?

positivity and negativity are qualitative values, not quantitative. What is the sqrt of '+' or '-'? These are actually FUNCTIONS, but functions which QUALify quantitative values. I didn't say there was no interrelation between quality and quantity - I was illustrating the limit of that relationship.

(3) Why do you think this expression is a problem?

OK, then solve for x
x=sqrt -1

 

[Thor]

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.

Has anyone considered the possibility that (sqrt -1) may indicate an error condition?

 

[turbo]

Has anyone considered the possibility that (sqrt -1) may indicate an error condition?

The field of complex numbers (numbers that chart in both real and imaginary numbers) gives rise to some pretty useful stuff. Sir Roger Penrose gives a little layman-level insight on complex numbers here:

http://www.princeton.edu/WebMedia/lectures/

Just scroll down to Oct 17, 2003.

 

[Hurkyl]

positivity and negativity are qualitative values, not quantitative. What is the sqrt of '+' or '-'?

I will agree that √+ and √- don't make sense.

But √(+1) does make sense, and so does √(-1) in the complex numbers.


"Negative" is a quality. "-1" is not a quality; it is an integer (or a rational number, or a real number, depending on the context -- the same symbol is used for each). "-" is neither a quality nor a quantity.

Just like "blue" is a quality, but "the blue pencil" is not a quality.


Incidentally, "-" does not "quality" a quantative value -- for any number x, there is no guarantee that -x is negative. (Though that is a common mistake)

 

[Nereid]

This is an interesting thread, but I wonder if it would feel happier in another part of PF - the philosophy of science and math, for example?

 

[Phobos]

No, no...I can feel a discussion about dark energy about to start at any moment. Hmm..it passed. Ok, off to PofS&M this goes...

 

[maverickmathematics]

I just wrote a really long and funny post but it really didnt express what i wanted to - what I really want to say is - yes maths is built on axioms that cannot be proven without the use of other axioms and so the entire syste, is underpinned by a created basis.

The original poster sid that the properties and theorems re all just built on a created base - well THAT IS THE POINT! You take 4 fundamental truths (peano's axioms) and then construct a number system, and then more and then more and then more! It is the purest of things to do - everything is done on sound deduction there is no faffing around with ifs and maybes - it is cut and dry 1 or 0 and that is the POINT of mathematics - when you prove something it is proved FORVER!

Argh i#m angry nad can't type ...i'll be back!

 

[Thor]

I will agree that √+ and √- don't make sense.

But √(+1) does make sense, and so does √(-1) in the complex numbers.


"Negative" is a quality. "-1" is not a quality; it is an integer (or a rational number, or a real number, depending on the context -- the same symbol is used for each). "-" is neither a quality nor a quantity.

Just like "blue" is a quality, but "the blue pencil" is not a quality.


Incidentally, "-" does not "quality" a quantative value -- for any number x, there is no guarantee that -x is negative. (Though that is a common mistake)

Yes - if x is, itself, a negative number then -x is positive.

HOWEVER, if you isolate and analyze each term in an equation, the properties of "+" and "-" are, indeed qualifiers. Math also breaches the domain of qualitative values when it uses ft and sec. . . ie 32ft/sec^2. "|1|" is a number "Distance" is not - "Time" is not. Those terms QUALify the numeric value. Numeric values are fungible - you can just as well use two 1's in place of 2. No matter how you term "ft" you will never get "sec's" - they are QUALitaively different.

 

[selfAdjoint]

Math also breaches the domain of qualitative values when it uses ft and sec. . . ie 32ft/sec^2.

Math doesn't do this. Physics does.

 

[Hurkyl]

The act of qualifying something is to take something general and add a qualifier to make it more specific.

That is precisely what "+" is not doing. "1" denotes exactly one real number -- it cannot be made any more specific.¹ "+1" means exactly the same thing as "1", so as such, adding the "+" has not made the symbol more specific: they're simply two different names for the same thing.

Similarly, "-1" is not more specific than "1"... in fact, "-1" is precisely something that "1" does not mean (unless you're doing arithmetic in characteristic 2). Adding the "-" changed the meaning of the string -- it did not make it more specific.

HOWEVER, if you isolate and analyze each term in an equation, the properties of "+" and "-" are, indeed qualifiers. Math also breaches the domain of qualitative values when it uses ft and sec. . . ie 32ft/sec^2. "|1|" is a number "Distance" is not - "Time" is not. Those terms QUALify the numeric value. Numeric values are fungible - you can just as well use two 1's in place of 2. No matter how you term "ft" you will never get "sec's" - they are QUALitaively different.

Doesn't this argument counter your thesis? It is a demonstration of being able to handle things you deem qualitative.


I disagree with your labelling, though. It is true that "1" and "1 ft" are qualitatively different -- for example, they're members of entirely different additive groups. But "ft" is not a qualifier -- "1" is not some vague thing that might be "1 ft" or "1 sec" or whatever -- "1" is simply "1". "1 ft" is something entirely different. Attaching the label "ft" changes the meaning of the symbol... it does not qualify it.

 

[metrictensor]

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.

I agree with what you are saying regarding the lliguisticcharacter of math and its limitations. All conceptual or systematic systems are limited by their axioms. Once the axioms are set certain results or emergent properties of these axioms are determined. [I am starting a post on this idea in particular and am looking forward to others' insights]

What he brings is a non-technical perspective that is very important. Too often those involved in the technical aspect of math lose sight of what is going on. They become too absorbed in proving and deriving and stop asking what they are actually doing. So to make the claim that just because someone is ignorant of the technical particulars their views should be dismissed an unfunfortunate missed opportunity. I can speak from both sides because I have a master in math but an not as expert as the Phds. I also like to look at it from a philosophical perspective with the formal training providing at least a basic background in the formalities. There is a danger from both sides. First from the technical side there is the danger of losing touch with the everyday. Math can, to a certain extent, be done in abstraction from the everyday. The fact that math can be done so abstractly opens the door for an over emphasis on this. The conventional and everyday-ness of math must be kept in mind.

Those who are not trained professionally run the risk of over-simplifying the technical aspect of math. I am not an expert here but the little I know of math makes me certain that math is a difficult subject and that mathematicians are crtainly intelligent. The rigors of doing technical math should not be overlooked.

I brought up these two points to emphasize that I am not trying to over simplify the efforts and acheivements of mathematicians by my talk of the everyday and conventional nature of math.

My philosophical view on math can be summed up this way. I proved that if a sphere is fit inside a cube such that all the sides touch the sphere then the ratio of the two shapes' volumes is equal to the ratio of the two shapes' surface areas. What did I do here? What do I mean by prove? How did I go about doing this proof? What prior work was mine based upon that I did not know of or took for granted? I see no difference in doing this problem than in doing a jig-saw puzzle. If I stare at my proof repeating to myself over and over the beauty of it am I any different than an artist who looks a their own painting in the same way? Both are perfectly acceptable in seeing beauty in their work. The difference is that the artist is well aware that they are the ones who created the painting. The mathematician on the other hand may look as if they have discovered something independent of their creation. This issue I call into question in my post on the nature of emergent properties.