Aspies For Freedom

B"H

Hello.  Thank you for reading my post.  I have taken Purim off from work, so I have a little bit of free time.  Actually, there is no work today.  Thus, I can explore a few "cool" areas such as mathematical philosophy.

I would like to delve a little bit in to mathematical philosophy again. I have not really studied it much in the past, so I approach it like a child would, with enthusiasm and a lack of formal training.  My main question for today, my "Purim Special," is as thus, "Why does Algebra work?" In essence, Algebra relies on the principle of homogeneity.  "X" can literally be any finite number, zero sometimes excepted.  

The same principles mostly work for all x.  X squared minus b squared is always going to be (x+b)(x-b), for instance.  Again, there are exceptions, but they are notable and not common.

So, if we go with a Nominalist explanation, it needs to be one that explains Algebra.  In fact, it would need to explain the homogeneity of Calculus as well.  [d(X^2)/dx]=2x.  This time it even works for zero.  And, unlike with Algebra, we can bring the "undefined", ie. the infinite, in to the matter.  We do not need to worry about what x is.  Algebra and Calculus permit a number system that is homogeneous and uniform.  Linear Algebra goes even further...And, beyond that I need a real education in number theory!

Now, Trigonometry might be considered an exception to this principle of homogeneity, as might Geometry.  However, even those sciences rest on homogenous principles.  Certain laws work, regardless.  And, much of the science behind both can be strangely connected to the circle as the n->infinity regular polygon.  What is going on here?

In fact, even mathematical Platonism is not an adequate enough explanation of this phenomenon.  We COULD conceive of a "real" mathematics that does not operate according to standard rules.   However, those rules do not seem to conform to the Universe in which we perceive our reality.  Mathematical Platonism can explain number sense.  However, Realism alone is almost as inadequate as Nominalism in one respect, that of explaining higher order mathematical thinking that is divorced from specific numbers.

So, what is going on here?  Please feel free to link to outside sources 'cause that's kind'a what I'm fishing for.

Personal belief: We are seeing a higher order design.

I look forward to yours.
All the best.

I'm not so sure that Algebra and all things arising from it rely on the homogeneity of variable input as much as define the variable as being any and every number. With the study of Calculus, this inclusiveness becomes especially apparent because there is no longer any need to confine X to being finite or to exclude zero. For example, as x->infinity in the equation y=(3x^2+2x+3)/(2x^2+3x-5), y->3/2. As x->-0 in the equation y=5/x, y->-infinity and as x->+0 in the equation y=5/x, y->+infinity.

I'm unsure of which exceptions your are speaking of with the general equation x^2-b^2=(x+b)(x-b). This equation is valid for all values of b. If b=1, then x^2-1=(x+1)(x-1). If b=-1, then x^2+1=(x+i)(x-i). If b=i, then x^2-i=(x+(isqrt(2)/2+sqrt(2)/2))(x-(isqrt(2)/2+sqrt(2)/2)). If b=-i, then x^2=(x+(-isqrt(2)/2+sqrt(2)/2))(x-(-isqrt(2)/2+sqrt(2)/2)). From there the solutions to every other value of b can be derived.

My education in advanced Number Theory is limited at best so I cannot comment too much here, but I would ask whether you could actually imagine a number system that is not homogeneous and uniform. I'm not even sure such a thing is possible.

Trigonometry and Geometry are really just extensions of Algebra to specific shapes. One could create a system similar to Trigonometry for any sort of polygon, but there would be little point to it as Triangles are more useful than other shapes and can also be applied to other polygons.

I'm really not sure such a mathematical system could be conceived. A system without rules is not a system at all.

I'm also don't think that abstraction is divorced from the reality so much that is a reflection of reality. If someone truly wanted to, they could graph all values for the equation y=x and it would take them forever because there are an infinite number of values. The equation is a representation of reality as well as an abstraction.

Sorry, all the things I would direct you to are in print form and I have been unable to locate them on the internet for your viewing.

I personally do not see a higher design in mathematics or anything else for that matter. My view of mathematics is that mathematics is a reflection of reality and reality is a reflection of mathematics. The two are inseparable. In this I suppose my views resemble the views of Pythagoras in that I believe reality is mathematical and that there could indeed exist one unified equation from which all others could be derived.

Mathematics appeals to me more than anything else in the world. It has become my passion, my obsession. I love the order of it, the logic, the perfection. It is the one true Science that rests above all the others.