Is Mathematics!Invented !

OR!Discovered?!

Camps!•  Platonists!•  Logicism (on the fence between these

two)!•  Formalists!•  Intuitionists / Constructivists!

Platonism!•  Math exists eternally and independent

of man.!

•  Mathematicians job is to uncover the relationships that already exist.!

•  Mainstream thought until “The Crisis”!

Platonists!•  Pythagoras (300 BCE) - Math is

religion. Religion is math.!•  Augustine (400 CE) - God created

math. Math reveals God.!•  Descartes (1625 CE) - Trust reason.

We can all find Truth.!•  Spinoza (1650 CE) - Reason is

supreme. The universe is a machine. !

Logicism!•  Mathematics exists as sets and logic.

All mathematics can be turned into these things (sets) and use this action (logic).!

•  Reaction to several developments in mathematics.!

Logicists!•  Bertrand Russell & Alfred North

Whitehead - wrote Prinicipia Mathematica!

•  Gottlob Frege - worked on recasting arithmetic as set theory!

Formalism!•  Mathematics consists of symbols or

expressions which are manipulated according to preassigned rules. Mathematics is not about anything it just is.!

•  Mathematicians go about their work by creating axioms and working out logical conclusions.!

Father of Formalism!•  David Hilbert (1862 - 1943)!•  Three steps!

– 1. Use formal language and formal rules!– 2. Develop combinatorial properties of this

formal language (agreed upon rules)!– 3. Prove using finite arguments that there

are not contradictions.!

Intuitionism / Constructivism!

•  The natural numbers are given by fundamental intuition.!

•  All mathematics should be constructed from this foundation.!

•  Constructions must occur in a finite number of steps.!

Intuitionists / Constructivists!

•  L. E. J. Brouwer (1908) - Dutch topologist!

•  Hermann Weyl!

The Crisis!

An Old Tale with a Twist:Little Red Riding Hood!

Cast:!Little Red Riding Hood - Platonist!

Wolf #1 - Antimones (Contradictions)!Grandma - Logicists!Wolf #2 - Brouwer!

Man with Ax - Hilbert!Mystery Stranger - Godel!

The Innocent Beginning!•  Little Red Riding Hood (LRRH) is traipsing

through the forest and happily finding all that is there.!

•  “The forest is so beautiful and wonderful. It must have always been here. I want to find out what I can about it.”!

Enter: Wolf #1!

Guest Appearance by Xena

Enter: Wolf #1!•  “Ah, you innocent thing, there are dangers

lurking in your forest.” (evil snarl)!•  LRRH gasps!•  “This is not the only forest to exist. You

believe a triangle has angles that add to 180º but there are triangles that do not. It is a very ugly triangle for this forest.”!

•  Pulls out ball and shows non-Euclidean geometry. !

•  LRRH is aghast. She puts her head in her hands and moans.!

Enter: Grandma!•  LRRH: Grandma, I thought you were home in

bed.!•  Grandma: No, LRRH, I knew this day would

come.!•  Grandma: My dear, we can fix this. Instead

of looking at the trees and the plants, look at a different level. Look at atoms (sets). We can make the whole forest out of it.!

Grandma and Wolf #1 Spar!Wolf #1: Oh, you foolish woman, what you

propose is . . . well, foolishness.!•  Your basic unit is a set. Okay. . . !•  R-set = set which includes itself!•  Ex: the set of all objects describable by

exactly eleven English words !(Yes, there are 11 words there)!•  So, the example is an R-set.!

Grandma and Wolf #1 Spar!•  Wolf: Now, think about another set. Let us

call it W-set.!•  W-set = the set whose members are all

possible sets except the R-sets!•  Ha - there are 11 words describing that set.!•  So, W-set (by definition) is not an R-set AND

W-set (by description ) is an R-set!•  You have a contradiction (evil snarl).!•  Enter Wolf #2 (unseen by Grandma and Wolf

#1)!

Wolf #2 Stirs the Pot!•  Wolf: Hello, LRRH. You think they are

making sense? Ha, they are talking nonsense. !

•  Wolf: There are no sets. There are only natural numbers. And, the only math there is comes from those numbers. Itʼs not beauty, itʼs brute force. (Wolf shows fangs in a smile.)!

•  LRRH: You are making my brain hurt. How can there be no beauty?!

Enter: Man with Ax!

Enter Man with Ax!•  Man: Here I come to save the day!!•  Man (to Wolf #2): You, fiend, you are

no friend of mathematics.!–  “If we would follow such a reform as the

one they suggest, we would run the risk of losing a great part of our most valuable treasure!” (Hilbert, C. Reid, p. 155)!

Wolf #2 Departure! Grandma: What you propose is

nonsense. !

Man: Yes, be gone with you.!

Wolf #1: Yeah, dog. There is only one wolf in this story.!

And then there remains. . .!Man: Grandma, you have tried to build some

sense for LRRH and restore her sense of beauty. I will help you with my trusty ax. It is called “formalism”. I will build (prove) all mathematics from a base and there will be no contradiction. (Looks pointedly at Wolf #1)!

•  Wolf #1: You know, I never did like axes. I think I will go find a rabbit to chase. (Exit Wolf #1)!

Enter: Mystery Stranger!

Enter: Mystery Stranger!•  Grandma and LRRH stand and watch Man

use ax to clear shrubbery. None see stranger enter. Stranger taps Man on Shoulder!

•  Stranger: Excuse me, sir. (Man turns around) I have the utmost respect for your work but I must point something out to you.!

•  Man: What can be so important? I am busy saving the world. !

Strangerʼs Revelation!•  Stranger: You are looking to build with no

contradictions. Well, I hate to tell you this, but, what you are building with is flawed.!

•  Man: What?!•  Stranger: My work shows that there will

ALWAYS be a contradiction when you start the way you have. !

•  Man: Then, my work will not stand.!•  Grandma: This is too much. Iʼm outta here.!•  (Stranger and Grandma leave in opposite

directions)!

Continuing the Journey!Man throws down ax in disgust.!LRRH: What am I to do?!Man: Do you still see the beauty of the forest?!LRRH: Yes. There is much here I would like to

know more about.!Man: I see that beauty too. Let us explore

together. If we are questioned by anybody, let me do the talking. !

LRRH: Men.!

The Journey today!“. . .[Mathematicians] are best described by

saying that we are Platonists on weekdays and formalists on Sunday. !

That is, when doing mathematics we are convinced that we are dealing with an objective reality whose properties we are attempting to determine. But then, when challenged to give a philosophical account of this reality, we find it easiest to pretend we do not believe in it at all.” (Cherowitzo, 2006)!

The End?!•  As LRRH and Man leaves, a Yeti pokes his

head around a tree.!•  Yeti: They all think they know what they are

doing. They are fools. They all are hooked on the same delusion - there has to be a foundation. All their foundations are full of holes. You can read it in “Proofs and Refutations”. . .if you dare.!

•  Dim lights.!

End Credits!Cast:!

Little Red Riding Hood - Platonist!Wolf #1 - Antimones (Contradictions)!

Grandma - Logicists!Wolf #2 - Brouwer!

Man with Ax - David Hilbert!Mystery Stranger - Kurt Godel!

Yeti - Imre Lakatos!

Sources!•  Cherowitzo, Bill (2006). “. . .the

Sequel” (powerpoint presentation)!•  Davis, P.J. & Hersh, R. (1981). The

Mathematical Experience. Boston: Houghton Mifflin Company.!

•  Hersh, R. (1999). What is Mathematics, Really?!

Thanks!!

Is it over?