Mathematics Describes Reality
Can you have a sheet of paper with only one side? Can you have a container with only an inside? The Möbius Strip and the Klein Bottle were inventions of topology, a study in mathematics that contravenes common sense. But they do exist; and they do have practical applications. As an investigation of relationships, topology is based on qualities, not quantities. Topology is nonetheless a study within mathematics. Topology is rigorous and consistent. It does not allow for internal contradictions, just as integer arithmetic does not.
My motivation here is a post reply from ewv who wrote: “Mathematics by itself doesn't describe reality. It is the means by which you relate in terms of concepts what can be measured. Mathematics is a science of method, not about things like physics does.” (Reply here https://www.galtsgulchonline.com/post... in "What is Science?" here https://www.galtsgulchonline.com/post...
Her keyboading error aside (“physics does” for “physics is”), she is usually a very adept student of Objectivism. As a quip, I once accused her of being Dr. Leonard Peikoff. Her comment about mathematics was a direct derivation of statements by Ayn Rand in Introduction to the Objectivist Epistemology, as well as elucidations by David Harriman in The Logical Leap. However, mathematics does describe reality, as does any language.
Imaginary Numbers Are Real. Pegasus is Not
http://necessaryfacts.blogspot.com/20...
My motivation here is a post reply from ewv who wrote: “Mathematics by itself doesn't describe reality. It is the means by which you relate in terms of concepts what can be measured. Mathematics is a science of method, not about things like physics does.” (Reply here https://www.galtsgulchonline.com/post... in "What is Science?" here https://www.galtsgulchonline.com/post...
Her keyboading error aside (“physics does” for “physics is”), she is usually a very adept student of Objectivism. As a quip, I once accused her of being Dr. Leonard Peikoff. Her comment about mathematics was a direct derivation of statements by Ayn Rand in Introduction to the Objectivist Epistemology, as well as elucidations by David Harriman in The Logical Leap. However, mathematics does describe reality, as does any language.
Imaginary Numbers Are Real. Pegasus is Not
http://necessaryfacts.blogspot.com/20...
Courant and Robbins' What is Mathematics describes topology as it evolved in the 19th century as about "the properties of geometrical figures [which are themselves abstractions] that persist even when the figures are subjected to [mathematical] deformations so drastic that all their metric and projective properties are lost", i.e., abstract properties that are invariant under continuous bijections. The metric properties of geometry are omitted as irrelevant in conceptualizing topological properties.
For a good introduction to the abstract nature of topology and its relation to metric spaces and mathematical analysis of a continuum, see the early chapters of the classic Introduction to Topology and Modern Analysis by George Simmons. It is so well written that it reads like a novel. http://www.amazon.com/Introduction-To...
For an introduction to how topological concepts are employed in the analysis of the abstractions of irrational numbers and the real number line see the first chapter of the standard Advanced Calculus by Creighton Buck, also very well written, like a good novel http://www.amazon.com/Advanced-Calcul...
Please do not make speculative assertions about who and what people are who you don't know and where they derived their knowledge from. In particular I am not a "she" and my knowledge of mathematics and its epistemology did not come from Dave Harriman's book, or from Ayn Rand's Introduction to Objective Epistemology alone, and you will not find it, especially the advanced ideas, in either of those books. The speculation only misleads those who read the assertions mistakenly believing that you know.
Being consistent and integrated mathematics in general or any of its special investigations can be approached many ways.
When we lived in Albuquerque, I worked as a substitute teacher in middle school. One day, I was given a special education class and the lesson plan called for The Pythagorean Theorem. These were kids who had trouble with basic arithmetic (or claimed to; some discussion there).
I did it the way the Egyptians did. We pushed all of the desks to the walls. I told them that the town had been flooded and we needed to redraw all of the property lines. Using lengths of yarn, we made 3-4-5 triangles. I had all day…
You can start with someone who cannot count and, beginning with trigonometry, teach whatever you need to.
I started the discussion of topology with the Moebius Strip and Klein Bottle as examples of "unreal" things that have reality - the twisted power belt that received a patent, for one. But, as you note, we could have started with the fact that the triangle, square, and circle are "simple" curves with an inside and outside and no boundaries crossing. A closed squiggle is another.
Neither Euclid's classic nor a modern high school textbook teaches geometry the way it developed historically. In this case, if I began with the Nile Floods and 3-4-5 triangles, you would reply that geometry is not about floods and rivers but about points, lines, surfaces, and volumes. It is not about measuring things like property markers but about abstractions of space. And, indeed, a university textbook in geometry will have no figures at all. But so what? Truth is real. Therefore, truths can be expressed many ways. That consistency is an attribute of truth, perhaps the sine qua non.
You asserted that mathematics does not describe reality "the way physics does." I point out that neither describe reality the way knitting does. Knitters - my wife's hobby, not mine - have a special way to describe textiles. It is factual, formal, and consistent. It allows the creation of new objects and new methods. My point here - and my assertion from the beginning - is that mathematics does describe reality or it would not work at all. More to the point (point… interesting concept….) mathematics offers ideas that seem unreal, such as "imaginary" numbers, but as we have seen, imaginary numbers are real.
And mathematics is a language.
And physics is a method.
A uber-level model that attempts to explain, well provide some explanatory mechanism for phenomena not yet understood is the mathematics giving the alternative universes 'model'. How such can be described as reality is beyond me, but, no reason mathematicians should not write sci-fi.
As for the square root of minus one, as this can not exist, to define it does not make it real. The concept is however a tool of great use in technology enabling calculations that would otherwise be far more tedious.
There is the the usual danger of reification - that because there is a word there must be something real that the word refers to. (Fairness, angels, carbon pollution, fiscal stimulus..)
One of my old favorite books comes to mind:
Mathematics, Queen and Servant of Science. Eric Temple Bell, 1952
The difference between mathematics and other languages is that you cannot invent a new "word" (concept) in mathematics unless it can be proved. The Greek goddess Nike has wings. They represent an idea, but they cannot be real. They are impossible to a human form.
Mathematics does not allow chimeras. Common language does, as you note: angels and fiscal stimulus, among too many.
You say that the square root of minus 1 cannot exist. But it does, as you agree, when considering alternating current electricity. What you mean - if I understand you correctly - is that we can create the square root of two by drawing an isosceles right triangle and taking the hypotenuse to any precision available. I agree that we cannot to do that to get the square root of a negative number.
The mathematics of global warming or the geography of heaven and hell with spheres of blessedness or condemnation or many other falsehoods are not the special fault of mathematics. The conclusion is just the point at which they stopped thinking. If they applied the mathematics (or common language) consistently and completely, the falsehoods would be revealed. Indeed, they are revealed, which is why we (you and I and others) reject them. Our success comes by holding common language to the standard of mathematics, i.e., to be logical.
(Thanks for the pointer to the Bell book. It is a classic that I have not yet read.)
Not sure I articulated that they way I could see it in my mind...challenging I must say.