Is Mathematics! Invented! OR! Discovered?!
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1 Is Mathematics! Invented! OR! Discovered?!
2 Platonists! Camps! Logicism (on the fence between these two)! Formalists! Intuitionists / Constructivists!
3 Platonism! Math exists eternally and independent of man.! Mathematicians job is to uncover the relationships that already exist.! Mainstream thought until The Crisis!
4 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.!
5 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.!
6 Logicists! Bertrand Russell & Alfred North Whitehead - wrote Prinicipia Mathematica! Gottlob Frege - worked on recasting arithmetic as set theory!
7 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.!
8 Father of Formalism! David Hilbert ( )! 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.!
9 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.!
10 Intuitionists / Constructivists! L. E. J. Brouwer (1908) - Dutch topologist! Hermann Weyl!
11 The Crisis!
12 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!
13 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.!
14 Enter: Wolf #1! Guest Appearance by Xena
15 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.!
16 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.!
17 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.!
18 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)!
19 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?!
20 Enter: Man with Ax!
21 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)!
22 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.!
23 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)!
24 Enter: Mystery Stranger!
25 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.!
26 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)!
27 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.!
28 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)!
29 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.!
30 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!
31 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?!
32 Thanks!! Is it over?
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