Post on 07-Mar-2018
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?