I have long been fascinated by math but I reached a precipice in school when I ran up against Calculus; I once asked my teacher if he could explain by example what in nature represents “the function of a number”—I was desperate, I said, because I could see the application of geometry and algebra, in which I excelled, but not calculus.
He looked at me disdainfully and said simply, “no.” That’s when I switched to Liberal Arts.
Ironically, my current fascination with computers is the result of their “living” functions—software—which are mathematical algorithms that perform tasks—they are active verbs – and of course the hardware/software relationship has been used as a powerful metaphor for the mind and brain.
But is it merely a metaphor or analogy? That is the essential issue of this post and the entire blog…
I Am a Strange Loop by Douglas R. Hofstadter is a wonderful book, a follow-up to a Pulitzer prize winning best-seller , Gödel, Escher, Bach: An Eternal Golden Braid that seeks to demonstrate the unique qualities of a mind that expresses itself in language, along with the inevitable gaps and paradoxes that result in believing too much in the logic of our spoken and written descriptions of “what is real.”
As a mathematician, neuroscientist and philosopher Hofstadter begins with the primacy of number because whatever symbols you use to represent “number”, certain truths persist.
For example, as Pythagoras famously asserted, the sums of the squares of a right triangle always add up to the square of the hypotenuse.
Remember – and consider – it does not matter what you call these concepts—they are mental constructs that are absolutely true.
Hofstadter, like Leonardo da Vinci, fucuses on a famous set of numbers that are also manifest in nature, and to many throughout history have represented a “Golden Mean” or perfect ratio, also called the number Phi (not Pi).
In mathematics, the Fibonacci numbers are the numbers that conform to this ratio in the following sequence of integers:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811
By definition, the first two numbers in the Fibonacci sequence are 0 and 1, and each subsequent number is the sum of the previous two. (in the relation of Phi).
If I understand Hofstadter’s key point, it is that the idea of such a sequence of number is primary and causal—and can be described in a different set of symbols, namely the English language as – a sequence of numbers such that each subsequent number is the sum of the previous two
Similarly one can define some members of this “set” of numbers in English as being “prime”, that is, indivisible by any number other than themselves and the number 1.
Okay, and there one can come up with very complex theorems and formulae to “describe” the relationships to ascertain which numbers, as one gets very large approaching infinity, are in fact prime and members of the Fibonacci sequence.
What Hofstadter points out, however, is the discovery of mathematician Kurt Gödel, that when one goes from the primary set of symbols (numbers) to our “understanding” of them represented by language; i.e., though about those symbols, very weird anomalies of logic come up that result in “strange loops”—infinite progressions without resolutions or perhaps paradoxes.
Still, in terms of scale, it is interesting to consider that there are vast Prime Numbers whose characteristics we can describe (indivisible by any number other than themselves and the number 1), and yet which our own brains and even the supercomputers we’ve invented have not yet “discovered”—yet which according to our language, analogies and suppositions must exist…
It was proven by Euclid that there are infinitely many prime numbers; thus, there is always a prime greater than the largest known prime (Wikipedia).
Here is a “prime” example of another such a paradox or anomaly:
The sentence “This sentence has ten words” has ten words. (I am a Strange Loop, page 140)
Here we can see how our verbal or linguistic description of a mathematical truth (which is absolute—see the infallibility of the Fibonacci sequence stretching out to infinity) is inevitably fraught with fallacy and “looping.”
This strikes me as significant to several levels. First if we look at how we use computer software to manifest concepts through software, we first write them out in code (language) and then compile them into a sequence of numbers (zeroes and ones) to express in “reality” (through the physical machine)—displaying on screen and interacting with other users.
To Hofstadter (I think), this paradoxical aspect of language is an obvious manifestation of mind which simulates nature on a very powerful level—by analogy it seems to mirror our own inner mental workings—but it cannot “explain” Nature or for that matter infinite sequences of number.
It can only explain characteristics.
Language, like our inner “I”, is looped and imperfect—with the inherent limitation of needing to be expressed in words, and consequently reducing the perfection of the absolute it describes (mathematical certainly; number) to what our limited minds can comprehend—fragmented, imperfect analogies to reality.
(Continued in Part 2)