"If anything seems certain, it is but the dead heart of the living tree." -William James
The Evolution of Math, from the Everyday to the Abstract.
It's difficult to appreciate in our highly mathematized contemporary world just how difficult it was to reach the levels of abstraction where math seems "natural" or obvious, or something that exists independent of human minds. Math develops out of actual human needs. What would be the meaning or use of "x to the fifth power"? Not until the 17th century (notably with Descartes) do we find even the use of consistent symbolic notiation for addition and multiplication.
DEVELOPMENT: Experience --> Science (useful abstraction) --> Math (symbolic reality considered without reference to use) --> ART (bringing the abstraction back to the human)
Origins of math-like abstraction: Needs of property and commerce: surveying/geometry, buying & selling.
Euclid's Elements: The ideal model of math as truth. Systematic treatment of knowledge originally gained through practice of Surveying.
Early (and long lasting) enthusiasm about the fundamental, universal nature of math:
Quoting Aristotle again ... "[the Pythagoreans] saw that the ... ratios of musical scales were expressible in numbers [and that] .. all things seemed to be modeled on numbers, and numbers seemed to be the first things in the whole of nature, they supposed the elements of number to be the elements of all things, and the whole heaven to be a musical scale and a number."
from Geometry in Art & Architecture: Unit 3
[The universe] cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word.
- Galileo, Opere Il Saggiatore p. 171. Source: Quotations by Galileo Galilei
We make the world in the form of math. Impress it's forms on matter, e.g., the equivalence of coca-cola
Math became the form everything has sought to cast itself into: e.g. Declaration of Independence
"We hold these truths to be self-evident, that all men are created equal"
Math as Cultural Legitimizer, the form of certainty.
The Myth of Equivalence.
Math begins as useful Representation, and becomes Self-Referential even considered to be Self-Subsisting.
"... mathematics is only the art of saying the same thing in different words"
- Bertrand Russell, Autobiography, Vol. 3, penultimate par.; source: Favorite Russell Quotations
In positing any equivalence, some thing(s) must be left out, eliminated in the reduction.
Let us still give special consideration to the formation of concepts. Every word immediately becomes a concept, inasmuch as it is not intended to serve as a reminder of the unique and wholly individualized original experience to which it owes its birth, but must at the same time fit innumerable, more or less similar cases—which means, strictly speaking, never equal—in other words, a lot of unequal cases. Every concept originates through our equating what is unequal. No leaf ever wholly equals another, and the concept "leaf" is formed through an arbitrary abstraction from these individual differences, through forgetting the distinctions; and now it gives rise to the idea that in nature there might be something besides the leaves which would be "leaf"—some kind of original form after which all leaves have been woven, marked, copied, colored, curled, and painted, but by unskilled hands, so that no copy turned out to be a correct, reliable, and faithful image of the original form.
Fred Nietzsche, On Truth and Lie in an Extra-Moral Sense
On Exactitude in Science
... In that Empire, the Art of cartography attained such Perfection that the map of a single Province occupied the entirety of a City, and the map of the Empire, the entirety of a Province. In time, those Unconscionable Maps no longer satisfied, and the Cartographers Guilds struck a Map of the Empire whose size was that of the Empire, and which coincided point for point with it. The following Generations, who were not so fond of the Study of Cartography as their Forebears had been, saw that that vast Map was Useless, and not without some Pitilessness was it, that they delivered it up to the Inclemencies of Sun and Winters. In the Deserts of the West, still today, there are Tattereed Ruins of that Map, inhabited by Animals and Beggars; in all the Land there is no other Relic of the Disciplines of Geography.
-Suarez Miranda, Viajes de varones prudentes, Libro IV, Cap. XLV, Lerida, 1658
(Jorge Luis Borges, Dream Tigers)
Quantization: The reduction of experience to numbers.
Quantification is extremely useful, but reductive.
Measure what is measurable, and make measurable what is not so.
- Galileo, Quoted in I Gordonand and S Sorkin, The Armchair Science Reader (New York 1959). Source: Quotations by Galileo Galilei
Real Numbers and Rounding Error.
Math + Media = Truth is in the perception.
The Crisis of Foundations.
Geometry, fell, not by being proved wrong, but by becoming one of many possible geometries, by the construction of alternatives, e.g., Lobachevsky's Imaginary geometry, which turns out to be the favored model in Einstein's universe (as quantified by Minkowski).
For God's sake, please give it up. Fear it no less than the sensual passion, because it, too, may take up all your time and deprive you of your health, peace of mind and happiness in life.
[János Bolyai's father Bolyai, Wolfgang (1775-1856) urging him to give up work on non-Euclidian geometry.]
In P. Davis and R. Hersh The Mathematical Experience, Boston: Houghton Mifflin Co., 1981, p. 220.
The quest for certainly turned to Arithmetic, and the most "obvious" thing, the Natural Numbers, counting. But, Number, which is so intuitively obvious (now), turns out to be highly problemmatic.
We used to think that if we knew one, we knew two, because one and one are two. We are finding that we must learn a great deal more about ‘and’.
- Eddington, Sir Arthur (1882-1944), In N. Rose Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.
The logic of Computers develops out of the quest to re-establish the certainty at the foundations of math, which lead through Arithmetic to increasingly abstract Symbolic Logic [Turing: computing machinery and intelligence - a.m. turing, 1950].
The fact that all Mathematics is Symbolic Logic is one of the greatest discoveries of our age; and when this fact has been established, the remainder of the principles of mathematics consists in the analysis of Symbolic Logic itself.
Bertrand Russell, Principles of Mathematics. 1903. source MacTutor: Quotations by Russell
I wanted certainty in the kind of way in which people want religious faith. I thought that certainty is more likely to be found in mathematics than elsewhere. But I discovered that many mathematical demonstrations, which my teachers expected me to accept, were full of fallacies, and that, if certainty were indeed discoverable in mathematics, it would be in a new field of mathematics, with more solid foundations than those that had hitherto been thought secure. But as the work proceeded, I was continually reminded of the fable about the elephant and the tortoise. having constructed an elephant upon which the mathematical world could rest, I found the elephant tottering, and proceeded to construct a tortoise to keep the elephant from falling. But the tortoise was no more secure than the elephant, and after some twenty years of very arduous toil, I came to the conclusion that there was nothing more that I could do in the way of making mathematical knowledge indubitable.
Bertrand Russell, Portraits from Memory. source MacTutor: Quotations by Russell
Pragmatic Salvation: Wittgenstein: math as USE => Math is a practice, as USED system; Axioms analytically determined.
"... what we call "counting" is an important part of our life's activities. Counting and calculating are not -- e.g. -- simply a pastime. Counting (and that means: counting like this) is a technique that is employed daily in the most various operations of our lives. And that is why we learn to count as we do: with endless practice, with merciless exactitued; that is why it is inexorably insisted that we shall say "tow" after "one", "three" after "two" and so on. -- "But is this counting only a use, then; isn't there also some truth corresponding to this sequence?" The truth is that counting has proved to pay. -- "Then\ do you want to say tha 'being true' means: being usable (or useful)?" -- No, no that; but that it can't be said of the series of natural numbers -- any more than of our language -- that it is true, but: that is is usable, and, above all, it is used.
Wittgenstein. Remarks on the Foundations of Mathematics, I-4
Not only rules, but also examples are needed for establish a practice. Our rules leave loop-holes open, and the practice has to speak for itself.
Wittgenstein, On Certainty [139]
When we first begin to believe anything, what we believe is not a single proposition, it is a whole system of propositions. (Light dawns gradually over the whole.)
Wittgenstein, On Certainty [141]
It is not single axioms that strike me as obvious, it is a system in which consequences and premises give one another mutual support.
ibid. [142]
Instead of NON-CONTRADICTION universal, we have the law of CONTRADICTION, or plurality of axiomatic systems.
PROGRAMMING & MATH
Math as making. => Given an arbitrary logical system, MAX, learn to use it.
BLACK BOXing: we can follow/construct a thing/theorem/object step by step, then we use it whole in larger constructions.
Each problem that I solved became a rule which served afterwards to solve other problems.
- Descartes Discours de la Méthode. 1637.
Everything in a computer is a math representation.
For us, the Proof of the pudding is in the tasting, the logic in the making: does it work?
For us, the goal is to make math meaningful, to bring it back to earth
Pure mathematics consists entirely of such asseverations as that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing... It's essential not to discuss whether the proposition is really true, and not to mention what the anything is of which it is supposed to be true... If our hypothesis is about anything and not about some one or more particular things, then our deductions constitute mathematics. Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.
- Bertrand Russell. 1872–1970, British philosopher, mathematician. Mysticism and Logic (1917) quoted in Proofs in Mathematics
Mathematics is a game played according to certain simple rules with meaningless marks on paper.
- David Hilbert (N. Rose, Mathematical Maxims and Minims, Raleigh N C, 1988) quoted in What is Math?
Mathematicians do not study objects, but relations between objects. Thus, they are free to replace some objects by others so long as the relations remain unchanged. content to them is irrelevant; they are interested in form only.
- Jules Henri Poincare (J. Fripp, M. Fripp, and D. Fripp, Speaking of Science, 2000) quoted in What is Math?
We want math AND meaning.
"Computers change the way mathematics is done, they change the way mathematics is taught and learned. Computers provide a vehicle for the evolution of an interactive diagrammatic method."
Proofs without Words on Cut-the-Knot.org
There is lots of math that the program can do, we will look at those in more depth, e.g., sound & sinusoids, image & matrices & 3D geometries.
back to: MATH + MEDIA