This is the missing link in my interview by Joe Johnston for www.perl.com. More details here
In Man and His Symbols psychologist Carl Jung said of numbers:
The very numbers you use in counting are more than you take them to be. They are at the same time mythological elements (for the Pythagoreans, they were even divine); but you are certainly unaware of this when you use numbers for a practical purpose.
Do you perceive "the mind of God" in mathematics? In other words, do you have occasional moments of transcendental awe in the course of your work? Or does your joy of math lie in the simple satisfaction of untying stubborn logical knots?
I think I can morph your question quite well into something I would fancy
What are the things mathematics studies? Numbers? "Numbers
and shapes"? Do we, mathematicians, discover the features
immanent to the
Universe, or do we invent them? Was the "idea of irrational numbers"
present-but-unnoticed before Egyptians (?) understood that sqrt(2) cannot
be rational, or did it jump into existence at this moment?
The discover/invent dichotomy is quite alive among mathematicians, I myself lean to the "discover" side. But is it important at the end, when you count what discoveries/inventions you can claim? More probably, it is like the tea-or-coffee choice: whatever makes you feel better now, is better for you. We may awe G*d creating things explainable in so beautiful unexpected ways, we may savor our ingenuity in inventing these explanations, but a joy remains a joy.
I believe the question about the "purpose" of math to be much more important. Obviously, it is not just "numbers and shapes", this was obsolete even in XIX century. Then what? Remember this lingering question in the title of Eugene Wigner's paper "The Unreasonable Effectiveness of Mathematics in the Natural Sciences"? Remember persistent attempt to apply math to more sublime, "non-natural" sciences? I do not remember miraculous results from these attempts. Should the application of math to medicine be confined to doing some statistical comparisons, or "clustering analysis" (whatever this means), or some other thing involving arithmetic manipulations? If you believe that the purpose of math is something else than investigating numbers, you may have a better chance to impact medical sciences. (I cannot miss an opportunity to mention my Teacher I. M. Gelfand's et al. book "On experience in collaboration of mathematicians and physicians".)
I do not think my stand on this is shared by a lot of people (though reading
"Schroedinger's cat" suggests that it should be similar to a part of
Bohr's Copenhagen program). First, the most juicy part: the math's next of kin
is poetry (well, maybe it is a twice or thrice remote cousin
;-), with linguistics coming close after.
Observation: breakthroughs in math (both pure and applied) come mostly from finding new points of view. (They may be either achievements in itself, such as the same explanation working for many different effects, or may set an old problem.) Math technicalities jump into existence to communicate these new points of view; additionally, time to time the progress comes via technicalities only - as in your "untying stubborn logical knots".
Then these technicalities become the focus of the new research, etc. etc. etc. For a layman, math studies technicalities via other technicalities. A professional can also see the hidden core made of new points of views behind these technicalities. But any sphere-of-mind enterprise is build by producing new points of view, so they alone cannot be the purpose of math. Technicalities should be taken into account too. How to do it without making math mundane-by-definition (which a lot of people know it cannot be)?
Let's focus on why technicalities jump into existence. Mathematicians use technicalities because we know that without them things are so hopelessly ambiguous that they become impossible to understand. The first step of the math training is "how to recognize ambiguities, so you can avoid them" (the skill which could revolutionize Usenet). The progression of rigorousness of math is due to discovering new ambiguities in what was thought before as unambiguous (as in: what is "the smallest positive integer which cannot be described in an English phrase with less than eighteen words"? Hint: count the number of words in the quoted part).
This singles out math by math communications going extra steps to be unambiguous. Emphasis is on the message producing the same effects on the receiving end, no matter who an (educated) reader is. But the related technicalities become the targets of the following math research. Ergo: the purpose of math is studying reproducible communications.
Remember what singles out natural sciences: they study phenomena which are (more or less) easy to isolate and reproduce on demand. Accepting the given above description of what math is, the enigma of Wigner's becomes a tautology: of course it pays to discuss reproducible phenomena in the language of "reproducible terms". By severely restricting your choices (your vocabulary etc.), you can actually achieve more, not less. Putting blinkers on your horse lets you get further quicker.
I know only one other intellectual enterprise which goes along the same lines, it is poetry. It also targets verbal communications of perceptions, and also improves the success of these communications by severe restrictions on the form of communications (of course, the restrictions are quite different). By the way: the best of poets achieve the level of reproducibility in their (nontrivial!) communications which rivals that of math. Math brings to masses these levels of clarity (achievable by a handful of people in a century), with a penalty: the topics of communication are restricted to those on the roads already paved by the progress of math.
Returning to your question: if you consider myphologemes as atoms the thoughts are made of, as the basic structures our mind organizes its perceptions into, as the primitive patterns bringing some method into the madness of our wordless consciousness, then numbers are without doubt myphologemes. So may be other concepts introduced by math. Math both extends the delicate toolkit made of myphologemes by offering the mind the new building blocks, and provides ways of condensing the high pressure vapours of our wordless concepts into the 1-dimensional streams of communications, freeing the space in the playground of the mind for new exciting developments.
And whether these developments discover islands of order in the otherwise chaotic Universe, or they bring order into the Universe, the power of excitement is still the same. The awe is transcendental either way.