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And Mathematics NTUITIONISM AND CERTANTY A Thesis Presented to The Faculty of Graduate Studies of The University of Guelph by JOHN WILSON TEMPLETON In partial fulfillment of requirements for the degree of Master of Science October, 1997 O John Wilson Templeton. 1997 National Library Bibliothèque nationale 191 ofCanada du Canada Acquisitions and Acquisitions et Bibliographic Services services bibliographiques 395 Wellington Street 395, nie Wellington OttawaON K1AûN4 OttawaON K1AON4 CaMda Canada The author has granted a non- L'auteur a accordé une licence non exclusive licence allowing the exclusive permettant à la National Library of Canada to Bibliothèque nationale du Canada de reproduce, loan, distribute or sell reproduire, prêter, distribuer ou copies of this thesis in microform, vendre des copies de cette thèse sous paper or electronic formats. la fome de microfiche/nlm, de reproduction sur papier ou sur format électronique. The author retains ownership of the L'auteur conserve la propriété du copwght in this thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor substantial extracts fkom it Ni la thèse ni des extraits substantiels may be printed or otherwise de celle-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son permission. autorisation. ABSTRACT NTUITIONISM AND CERTAINTY John Templeton Advisor: University of Guelph, 1997 Dr. Murray Code This thesis is an examination of L. E. J. Brouwer's Intuitionism. especially as this philosophy attempts to get at the meaning of "certain truth" in mathematics. In the intuitionist view mathematical certainty rests solely on the self-evidence of intuited objects and constructions which are sense-perception independent. Construction takes place mentally in a pure intuition of time which is believed to be a permanent feature of the human mind. Objects which cannot be intuited or constnicted do not exist. If Brouwer is correct in his view of mathematics, and mathematical truth, this implies that there are no non-experienced truths. This position, however. has been somewhat weakened by the work of Turing, who has shown that there may exist mathematical truths which are independent of construction. The existence of tmths independent of the best efforts of mathematicians is also suggested in mathematical platonism. Platonism on its own, however, seems incapable of providing a means of getting at any of these truths, thus rendering it useless to provide any certainty. If, however, intuitionist construction is joined with the basic tenet of mathematical platonism, we might then be able to consider experienced, or constructed. truths as certain. Ackno wfedgmen~ It is said that no one is an island, and I am certaidy no exception to that. 1 am very grateful for the assistance I received throughout this work. To the department of Mathematics and Statistics at the University of Guelph I give thanks for simply allowing me to pursue an advanced degree in an area which interests me. Thanks also are due my advisory committee, but especially Dr. Murray Code. 1 enjoyed very much our philosophical discussions, and have corne to greatly appreciate Dr. Code's wit and sensibilities. 1 must also acknowledge the support my God gave me as 1 struggled through this work. but especially the support I found through the death of my father in the second year of my studies. 1 am disappointed that my dad couldn't see this work. but I am very glad he was my dad. To him 1 dedicate rny thesis. CHAPTER ONE UNCERTAIN CERTAINTY .......................................... 1 1 . A view of certainty ........................................ 1 2 . Foundational woes ........................................ -6 3 . Mathematics and philosophy ................................ -8 CHAPTER TWO THE A PRlORl AND UATHEUA TICS ................................13 1. Synthetic a priori propositions ............................... 13 2 . Going Beyond the Concepts ................................. 16 3 . Mathematics teaches Metaphysics ............................ 17 4 . To perceive. or not to perceive ............................... 19 5 . The fom of perception ....................................71 CHAPTER 3 CERTAIN CONSTR UCTIONS? ...................................... 25 1 . Building on intuition ...................................... 25 2 . Brouwer's intuitionism in brief .............................. 26 3 . Constmcting knowledge? ................................... 30 4 . Construction versus language .............................. -233- CHAPTER FOUR DO COMP UTERS HA VE INTUITION? .............................. -38 1 . Necessary constnictions ....................................38 2 . Construction as computation ................................ 41 3 . Compressions and the uncornputable .......................... 43 CHAPTER FIVE A CERTAIN CERTAlNTY .......................................... 50 l.InorOut? ............................................... 50 2 . Mathematics "out there"? .................................-54 3 . Making Contact ..........................................56 4 . Intuitionism and mathematical platonism mamied ................ 57 REFERENCES ........................................................61 CHAPTER ONE UNCERTAIN CERTAINTY Several yean have now passed since I first realized how rnany were the false opinions that in my youth 1 took to be tue. and thus how doubtful were ail things that 1 subsequently built upon these opinions. From the time 1 becarne aware of this, 1 realized that for once 1 had to raze everything in my life, domto the very bottom, so as to begin again from the first foundations... Rene Descartes' 1. A view of certainty We al1 want to be able to trust that some aspects of our Iives are in some sense secure. It's rather important, for instance, to know that the sun will rise tomorrow. We like to say that Our friendships are enduring, and that we can count on our friends. Life sometimes seems a mass of ever changing confusion, with accidents, death, war and poverty inundating the evening news. There cornes a certain steadiness in knowing that we can rely on at least some things to remain sure. Afterall, if there were nothing, real or imagined, which can be tnisted for stability in our lives, chaos is al1 we cm count on. But what is it to be able to trust, to count on something? It is to rest secure in the knowledge that tomorrow the sun will corne up, and that our friends remain loyal. These exarnples are somewhat misleading, however since fnendships do fade, and physicists tell us that 'Rene Descartes, Meditations on Fint Philoso~hv,tram. Donald A. Cress (Indianapolis: Hacket Publishing Company Inc., 1979), 13. one day the sun won? rise but will rather engulf the earth as a red giant. And yet the desire for stability, for certainty in life, remains. A natural question to ask then is whether there can exist, and if so in what way, any knowledge which is certain, anything which can be counted upon as being tnie without qualification or question. Another way to ask this question is to wonder whether there is any knowledge in the world which is so certain that it would be unreasonable to doubt it. Questions about the nature of knowledge and certainty are some of the most difficult in philosophy. How do we know things? What makes knowledge sure? What is midi? Perhaps thousands of volumes have been written attempting to get some clarity on issues such as these, and yet it seems unlikely that the definitive answer can be found in any one of them. It may be that we can never have complete certainty that what we "know" is the "û-uth" (although it does seem odd that if this were the case that we could ever be certain of that!). This thesis will not provide definite answers to these questions either, but will rather be an examination of one particular idea about knowing, namely the mathematical intuitionism of L. E. J. Brouwer. Before proceeding with Brouwer, though, I would like to begin with a brief overview of Antonio Rosmini's ideas surrounding intuition and certainty in order to provide a jumping off point for our discussions of mathematicai intuitionism. Knowledge of anything, I will assume, requires sornething about which we can have knowledge. Knowledge of some object, Say a mathematical proposition, need not be certain knowledge, however. We may be, for instance, mistaken in ou.understanding of the 'rneaning' of that proposition. Thus what we take to be "tnie'of the proposition, and what actudly is true of the proposition rnay diverge.' Does mistaken understanding constitute uncertainty? No, for 1 rnay be firmly convinced of my belief that the proposition means exactly what I conceive it to mean even though at a later date it can be shown to me that the belief 1 held at the time was indeed false. The nineteenth century Italian philosopher Antonio Rosmini defînes certainty as "a fm and reasonable persuasion that conforms to the tr~th."~To be able to be persuaded carries the implication that assent cm be given or withheld. Rosmini writes that "anythng whatsoever to which we give or withhold our assent can be expressed as a proposition'*. meaning that in his view our knowledge consists in propositions. For our present purposes I will define a proposition as any statement which has a subject and a predicate, such as "granite is a hard stone". To be uncertain does not rnean that one is without an opinion or belief about some proposition. It does mean for Rosmini5, though, that one is not convinced that the belief held does indeed conform to the tnith. Thus,
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