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
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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, for Rosmini, if one's knowledge of some proposition is to be certain three things are required. First, the proposition must have, or be capable of having, 'tmth' attached to it. Secondly, if one is to be "certain" in their knowledge of that proposition, they must be convinced that their
'A mathematical proposition can be of the form x < 3, where the 'truth set' of the proposition are those elements of the domain which make the statement true. For instance x = 2 belongs to the tmth set, whereas x = 4 does not.
'Antonio Rosmini, Certainty, tram. Denis Cleary and Terence Watson (Durham: Rosmini House, 199 1), 4. belief conforms to the truth, and finally that conviction mut have a reason or motive6 behind it. If, however, we can be mistaken in our convictions about the truth of sorne proposition, how cm we ever be 'certain' that our "firm and reasonable persuasion" does indeed "conform to the tnith"?
Rosmini distinguishes between two principles of certainty which he calls
. . 'œintrinsic'7and "extrinsic". Describing them he writes:
The first is directed not only to persuading and convincing us that tmth must be present in the proposition, but it is part of the proposition, showing us its tmth clearly and making us intuit it with the eyes of our intellect. On the other hand, the second is not in the proposition, nor does it always relate to its contents ...we prove to ounelves by means of the principle and become rationally convinced that the proposition must contain the tnith. We rnust therefore give our full assent to its contents.'
By this notion of being able to "intuit with the eyes of our intellect" Rosmini means that the truth of a proposition is irnmediately apparent to our mind without the need of any mediating reason (or what he calls a "sure sign of truth") which would persuade us of the truth because, as he writes, "truth seen directly by the mind through intuition.. .is self-evident, devoid of sign. and without intervening arg~rnents.''~When such a proposition is presented to our mind for consideration we just see the tnith of it, and so the "reason" which persuades us of the proposition's truth is contained in the proposition itself. This is the principle of intrinsic certainty. For those propositions, the tmth of which our minds cannot know directly, we need something to 'assure' us of their
'Rosmini, Certainty, 5.
'Ibid.. 1 1.
'Ibid., 12. truth. But when we do have that assurance, that "sure sign", we can then be certain of the truth of the proposition by the principle of extrinsic certainty. The sure sign of truth, extemal to the proposition under consideration, provides the "reason" which persuades us of the truth. Of course, as Rosmini rightly points out, sign, in order to be effective, must first be certain itself.'" Now, if the sure sign is itself dependent upon anodier sure sign for us to be certain of the first, and if the second is again dependent on a third, and so on, we see the potential danger of a infinite regress. Rosmini attempts to get around this problem, however, by insisting that at some point the "sure signs" must end in a tmth known in itself.1° If this be the case, then the knowledge of which we can be certain is ultimately only t-hat which is either intuitively known, or that which issues forth from an intuitively known truth in a series of these b'sure signs". In other words, for Rosmini knowledge of which we can be certain somehow originates in the intellect through a direct "intuition" of the "truth".
The mathematician L. E. J. Brouwer is another who also believes that human knowledge somehow originates in mental intuition. Brouwer writes that "proper to man is a faculty which accompanies al1 his interactions with nature, namely the faculty of taking a mathematical view of his life, of observing in the world repetitions of sequences of events, i.e. of causal systems in tirne ...these sequences thereupon concentrate in the intellect into mathematical sequences. not sensed but observed"" Just how is this mental
'*observation" Brouwer speaks of possible? Does he mean to imply that what we can only
be sure of is mathematical knowledge, or does he mean that knowledge is somehow
mathematical? These questions will be given some consideration in this thesis, but the
central question I wish to examine is whether Brouwer's "intuitionism" might be able to
provide any knowledge of which we can indeed be "certain". Brouwer's thinking arose in
a time when there was a perceived crisis in the foundations of mathematical knowledge.
2. Foundational woes
In the very early stages of this cenhiry it was discovered, independently by
Bertrand Russell and E. Zermelo, that the logic in Cantorian set theory gave rise to
contradictions. At the time that Russell's paradox, as it came to be known, surfaced.
Russell wrote a letter detailing the paradox to the great logician Gottlob Frege who was at
the time finishing the second volume of his work on the logical foundations of arithmetic.
Frege's response to Russell's news appears as a dispirited acknowledgement in his book.
Frege writes:
A scientist can hardly meet with anything more undesirable than to have the foundation give way just as the work is finished. In this position I was put by a letter fiom Mr. Bertrand Russell as the work was nearly through the press."
What is this paradox that gives Frege cause to believe that the foundation he has
"L. E. J. Brouwer, "On the Foundations of Mathematics" Brouwer: Collected Works, ed. Arend Heyting (New York: North Holland Publishing Company, 1975), 153.
I2~ottlobFrege, quoted in, Pi in the Sh,John Barrow, (London: Penguin Books, 19931, p. 111. tried to give arithmetic fails in its light? To begin understanding it, consider the finite set of al1 rubber balls. It is clear what objects belong to this set, and which do not. Finite sets are wondefil to deal with since they can be actually inspected, at least in principle, to see whether the contents of the set conform to the charactenstics of the set, providing hope that a limited type of certainty may be found here, in mathematics, after ail. Infinite sets however are not at al1 easy to check. It is impossible to inspect an infinite number of members of a set to determine whether they are "true" to the set's characteristics.
However we can at the least understand the nature of the eIements of an infinite set.
Consider. for instance, the set of al1 even natural nurnbers. While one cannot directly verie whether each and every element in the set has the requisite property of the set in which it's situated, one can nonetheless be "certalli" of what can belong to the set. For instance, by definition of "even natural number" 1 am "certain" of the tmth of the proposition "3 cannot belong to the set of even naturals, whereas 798 036 cm."
Consider next the Set of al1 sets which are not mernbers of themselves. Herein lies Russell's paradox, for we can ask whether that set is a member of itself or not. If the
Set of al1 sets which do not contain themselves as members does not contain itself, it must then by definition contain itself, which is a contradiction. Altematively, if it does contain itsetf as a member then it cannot belong to the Set of al1 sets which do not contain themselves as members, which is once again a contradiction. It may be that the root problem of this paradox which so troubled Frege and Russell, is the fact that the Set of al1 sets is selfreferencing. Whatever the case, Russell's paradox, and others like it gave rnathematicians cause to attempt to banish antirnonies from mathematics, perhaps with the hope of forever rendering mathematical knowledge somehow certain. Much in those attempts was philosophical in nature.
3. Mathematics and philosophy
When rnathematicians and philosophers began turning their attention to the problem of the foundations of mathematics it sternmed in part, says Dummet, from
Frege's forcing "both philosophers and mathematicians to acknowledge the lack of any satisfactory philosophical account of the nature and epistemological basis of mathematics."" Mathematics was without a fondation, and that state was regarded as unacceptable. In this century there arose three main approaches to the problem of the foundation of mathematics: Logicism, Formalism, and intuitionism. Frege, Russell and
Whitehead are generally considered the strongest proponents of Logicism. Formalism is associated with the narne David Hilbert and his idea that mathematics is the study of forma1 axiomatic systems and the results arising fiom those systems. Finally, as 1 have mentioned, Intuitionism owes it's beginnings to the thinking of Brouwer.
Both Frege and Hilbert believe that mathematics needs sorne kind of foundation to give it's results epistemological justification. Frege attempted to demonstrate that through an appeal to the fundamentai principles of logic the basic principles of mathematics could be derived. Mathernatics, for Frege, would therefore be directly justified by logic. Hilbert, on the other hand, wished to show that mathematics is an internally consistent system in itself which can be used to answer any conceivable
I3Michael Dummet, Elements of Intuitionisrn (Oxford: Clarendon Press, 1977), 1.
8 mathematical question. Simply state the question in a mathematically meaninghl way, run it dirough the formai system, and the answer will fa11 out. Hilbert believed that if the methods used in mathematics are "certain", then the certainty of the results of classical mathematics will foilow. Hilbert writes that "the goal of my theory is to establish once and for al1 the certitude of mathematicai methods."'" Brouwer, on the other hand, rejected the notion that classical mathematics, as it was developed by Cantor, et al, (with its acceptance of actual infinities) could be given a secure foundation. Brouwer believes that rnathematics when properly piacticed, should not need any extemal justification, but should "wear its own justification on its face.'"' Since classical mathematics did not have this quality, the intuitionists hold that mathematics needs to be reconstxucted fiorn sound principles rather than provided with a justification dubiously constmcted 'afier the fact'.
In the intuitionist view, proper mathematics consists in the "constniction" of objects in the "pure intuition of tirne""! Only those mathematicai entities which have been consrructed exist. for the intuitionist. as opposed to pure existence theorems, for example. which might at best only point to a possible construction but which is without any means of any actual construction. Logico-linguistic arguments are not mathematics for the intuitionist, nor do they provide any kind of justification to mathematics.
'"David Hilbert, "On the Infinite" in Philosophy of Mathematics: Selected Readings 2" -Ed., Putnam and Benacerraf Eds. (Cambridge: Cambridge University Press, 1983), p. 184.
"Michael Durnmet, Ibid., p. 2.
"%tephan Komer, The Philosophv of Mathematics: An Introductory Essay, (New York: Dover Publications hc.,1986), p. It is generally acknowledged that Logicism and Formaiisrn, in their original intents, are largely considered to be failed programmes; Logicism because of its inability to ameliorate the paradoxes of set theory; and Formalism because of Godel's incompleteness theorems." Nonetheless logic continues to remain vital to the mathematical enterprise, and many mathematicians seem to pursue mathematics as though they were confessed formalists. According to Dummet. however, intuitionism "is the ody unified system which survives intact fiom the earlier period when rival philosophies of mathematics were in conflict, each claiming to hold the key which would dockal1 the doors" ''to certain mathematical knowledge. But what might "certain" mathematical knowledge consist of! It may seem unquestionable that we cm be certain of the mathematicai proposition "1 + 1 = 2", but nonetheless one can wonder why one can seem "certain" of the truth of this proposition. 1s certainty assured because of the way the proposition carries the information, or does the certainty reside in the manner in which the proposition was constnicted? These are not easily answered questions, but Brouwer's intuitionism, as we shall see, may provide some possible answers.
The word intuitionism conjures up notions of innateness, and knowing without study or thought. To have an intuition is sometimes considered to be in possession of a mysticai 'other' sense which provides knowledge cornpletely mattainable for those without it. But Brouwer and his disciples do not imply by the narne given their philosophical view that the mathematician need not exert any effort in understanding
"Michael Durnmet, Elements of Intuitionism, 2.
"Ibid., 1.
1 O mathematics. Most students of mathematics would deny this picture of effortiess intuition
as being applicable to mathematics. Studying mathematics is hard work. On the other
hand, some might argue that mathematics does indeed deserve the narne intuitionism as it
is a peculiar reairn of thought foreign to much of humankind. Mathematics holds
knowledge inaccessible to none but the initiate. or so is sometimes said. The rneaning of
mathematical intuition in Brouwer's philosophy, however, is much more subtle.
Mathematics, for Brouwer naturaily arises sirnply because we hurnans think.'9In
a sense an intuitionist might Say "we think, therefore mathematics is" to borrow Frorn
Descartes' farnous "1 think, therefore 1 am". This calls for clarification. Intuitionism is not a platonic philosophy (at least not intentionally so). Mathematical reality, for the intuitionist is not "out there" existing extemal to our minds waiting to be discovered.
Rather, as one properly uses his or her intuition, mathematicai reality can be constructed in the mind. Unfortunately for the intuitionist, however, there still remain some deep metaphysical problems with such a conception of mathematical reality. For example. attributing mathematical existence only to those things constnicted provides no account for the intuition of ideas that presumably motivate the direction of some mathematical constructions. If an idea seems to be correct, Say, through a completed construction, one may ask when the "constnicted" mathematical entity began to exist. Does the motivating idea point to an entity extemal to the mind of the mathematician after dl, in a platonic sense? Another problem seemingly inherent to intuitionism is the very notion of intuition.
19 L. E. J. Brouwer, "Intuitionism and Formaiism" in Philosophv of Mathernatics: Selected Readings 2"Ed., Putnam and Benacerraf Eds. (Cambridge: Cambridge University Press, 1983), p. 80 How can mathematicai "tmths" be apprehended directly by the mind? With these considerations in mind, it is therefore important to ask Brouwer "what is exactly your conception of intuitive construction, and what sort of objects can be given through it?"
Since Brouwer's idea of mathematical intuition is essentially that of Immanuel Kant, it would be well to tum to km first. CHAPTER TWO
THE A PRIORI AND MATHEMATICS
Human reason has a natural propensity to transgress the limit of experience" Mathematics is a splendid example of pure reason's self-expansion without the aid of experience." Immanuel Kant
1. Syntheîic a priori propositions
Kant calls a proposition (a statement asserted by somebody) a judgment, and it is these (personal) assertions in which Kant is interested. Kant classifies judgments as either analytic or synthetic because he thinks that within a judgment, the subject and predicate can be related to each other in one of two ~ays.~'Either the predicate is contained in the subject, or it lies outside the subject, but is still connected to it in some way. For instance, to negate the judgment that 'green is a colour' leads to a contradiction. Colour is
"contained" in green. Such a judgment is called, by Kant, analytic." The distinguishing
'Ohmanuel Kant, The Critique of Pure Reason, trans. Wolfgang Schwarz (Darmstadt: Scientia Verlag Aden, 1ND), 205.
"Ibid., 224. criterion for analytic judgments is that they cannot be negated without arriving at a
contradiction in terms. Thus analytic judgments can be considered "trUe7' by vheof
their structure." It must also be noted that an analytic proposition conveys information
dealing with the meaning of terms only. No secondary information derived frorn sense or
expenence is provided. On the other hand in the judgment that a rainy day is a cold day
the predicate 'cold day' is not contained in the subject 'rainy day' and yet the predicate
does provide us with Merinformation about the subject. Also, to negate this judgment
does not necessarily lead to a contradiction. These sorts of judgrnents are called by Kant
synthetic.
To his classes of analytic and synthetic judgments Kant adds a Mer
classification, for judgrnents may be either apriori, or apo~teriori.'~An a priori judgment is logicdly independent of olZ judgrnents which descnbe experience and sense
impressions. Two judgments cm be said to be Iogically independent if neither
necessitates the other or its negation; and if the same is true also of their contradictones.
For instance. the judgments "my car is blue", and "my car is a FireFly?'are logically
independent for one does not necessitate the other. The judgments "my car is a FireFly",
and "my car is a Pontiac" are, however, logicaily dependent, for only Pontiac makes the
FireFly, and so the first judgment implies the second. Now the judgrnents "my car is
blue" and "my car is a FireFly7', while logically independent, are not a prion judgrnents
'?hilip Kitcher, "Kant and Mathematics" Kant's Philosophy of Mathematics, ed. Car1 J. Posy (Boston: Kluwer Academic Publishen, 1992), 1 10.
"Stephan Komer, Kant (Baltimore : Penguin Books Inc., 1955), 20. for they descnbe sense impressions. Examples of a priori judgments are 4 + 8 = 12, every
triangle with two equal sides is necessarily isosceles, and every father is necessarily maie.
While it is txue that one's knowledge of these judgments does depend in a way on
expenence, that dependence is not what is meant by logical dependen~e.~'For instance.
the meanings of the concepts "father" and "male" may corne to me in the course of
growing up, but 1 cmnonetheless "know" that every father is necessarily maie wiîh out
having to ask every father on planet earth to drop his pants, so to speak. nie judgment is
"true", by virtue of the meanings of the concepts in the judgment, independently of my
experience. An a posteriori judgment is one which is not a priori, meaning that an a
posteriori judgment is dependent on other judgments which describe expenences or sense
impressions.
With the classifications analytic, synthetic, a priori and a postenori we have four
possible types of judgments: analytic a priori, analytic a posteriori, synthetic a priori, and
synthetic a posteriori. Of the four it is readily apparent that analytic a posteriori judgments
are not possible since this would be a contradiction in terms. An analytic judgment does
not provide the type of information that an a posteriori judgment does. Of the three
remaining possibilities it is clear that al1 analytic judgrnents are a priori since an analytic judgment, in conveying information about meanings of tems only, is independent of expenence and sense impression. Thus, since al1 analytic judgments are a priori, a
posteriori propositions must be synthetic. The remaining possibility, synthetic a priori judgments, are those judgments whose predicates lie outside their subjects and yet which are logically independent of ail judgrnents describing sense-experience, and it is this sort of judgment which Kant believes metaphysics, when properly done, should contain. He wants a metaphysics which expands understanding by the use of a priori propositions which add to a concept something not contained in it. It is Kant's fm belief that
"synthetic judgments a priori, even transcend experience"".
2. Going Beyond the Concepts
Synthetic a priori judgrnents are, for Kant, fundamental to the natural sciences, and in particular to ma the ma tic^.'^ An example will illustrate of the nature of a synthetic a priori judgment. Consider the mathematical judgment 7 + 5 = 12. Most would agree that this statement is a priori, but Kant would fuaher argue that it is also synthetic. in order to get 12 from 7 + 5 a procedure of addition must take place. That is to Say that 12 is not contained in 7 + 5, in the analytic sense. Kant writes that "in the concept of a sum of 7 and 5 [we] have only thought the concept of a unification of two numbers, but not the sole number that comprises both. Intuition must be called in, and we must go beyond the conceptual content to obtain the an~wer."'~By intuition Kant has two ideas in mind; narnely a sort of "perception", and the form of that "perception". An intuition takes place when an object "affects the mind in a certain way."" The ability that we have to be
"~ant,Critique, 1 1.
%id.
291bid.,1 1-
"Ibid., 16. affected by objects is called by Kant "sensibility", and it is our sensibility which supplies intuitions. Sensation is the actual effect of an object on the "faculty of presentation"
(sensibility). If the intuition contains no objects, but only relations between potential objects, it is cailed the form of int~ition.~'The form of an intuition can be compared to a box inside our mind in which or sensibility is afFected. Now, an intuition of the concept of the union of the two numbers 7 and 5 need not contain the concept 12 prior to constmcting the concept 7 + 5. The concept 12 is 'perceived' after the initial construction of the 7 + 5 and, in a sense, one's knowledge is expanded. This expansion of knowledge is a priori in that the concept 12 was not perceived in sensory experience, but is rather independent of any sense (save that of intuition). Knowledge has been built by reason unaided by empincal data, and we "know" by mental inspection of the concepts. We "go beyond" the conceptual content, and for Kant this is a key idea, especiaily in light of his desire to establish what is proper in metaphysical reasoning.
3. Mathematics teaches Metaphysics
According to Kant, mathematics has advanced so well because it is a science in which it's objects of inquiry are presented to reason a priori. And an a priori perception is called pure, by Kant, if it is completely fiee from sensory experience." That is, an a priori
'perception' is one which is made by reason alone via intuition. But how is this accomplished?
' Ibid.
3'Ibid., 17. Consider, for instance, the a priori perception of a triangle. The properties of the triangle are not understood by sensibly perceiving the triangle with its properties and then studying those perceptions (much in the way a tree might be studied). Rather the properties of the triangle are understood as the triangle is produced, or constructed,
"according to concepts, thought into the figure and exhibited by construction a pri~ri."'~ ui a way, the triangle is known as it is constructed, and is not known pnor to that construction.
Kant believes that metaphysical reasoning should proceed dong a similar line as the 'objects' of its inquiry are not of the sort which are sensibly perceived. Freedom, for example, is not an entity which makes itself available to my eyesight, or any other physical sense. Rather, if 1 am to know the concept fieedom it must be 'perceived' by my reason. By a perception of reason (that is a perception a priori) Kant means a perception which "shall establish something about objects before they are given to us [sensibly]."" It is Kant's contention that this sort of perceiving involves assuming that an object perceived a priori will conform to the intuition rather than the perception conforming to the object. He gives the example of Copemicus' celestial insight that we earthlings are not the center of the universe, that in fact it is we who move through the heavens rather than the stars orbiting us. Copemicus' insight reshaped our view of reality, and it is generaily agreed that he was right. The actual structure of the cosmos did conform to his
"intuition" rather than to what was expenenced via the sensible apparatus of perception we cal1 sight. It is these pure perceptions of reason which Kant thinks comprise the whole of mathematics and, if so, ment our attention. Yet, what does it mean to perceive without sense?
4. To perceive, or not to perceive
When one has perceived something, a judgment can then be made about that perception. Kant rnakes a distinction between the two activities of perception and judgment making, since judgment seerns to presuppose perception. Kant likens perceiving and judging to two different capabilities of the minci; sense and understanding.
Kant writes that "the capacity (receptivity) for obtaining presentations according as we are af3ected by objects is called sensibility. By means of sensibility, therefore, objects are given to us, and it alone fumishes intuitions; [objects] are thought, however, by the understanding, and fiom it arise concepts."35 Using ow understanding, Kant argues, we make judgments about what we previously perceived through sense. Without perceptions. judgments would not be possible, for we must make judgments about somerhing. ludgments, then refer to that which is perceived.
Kant calls those things about which we can make judgments, objects. In making an empirical judgment about some sensible object such as a coffee rnug, for example, we employ concepts abstracted from previous perceptions. For example, in making a judgment such as "my coffee mug is red" the concept of 'redness' is applied to my particular mug before me, and that judgment describes (at ieast in part) what my mug is like. The general notion of redness was previously abstracted fiom other sensible objects which contained a perceivable similar quaiity, or character 1 label generally as 'red', The concept 'red' is now available to rny mind for application in making judgments about various different objects which might, or might not contain this concept of redness. 1 can thus "know" that rny mug is "red". Kant cails these sorts of concepts, which are abstractions fiom sensible perception, a posteriori concepts. in applying an a posteriori concept to some sensible object, an a postenori judgment is made. It would seem then, in keeping with Kant's system, that there must exist a priori concepts, such as the concept of a union of the mathematical concepts 7 and 5, which when applied allow one to make an a priori judgment. Kant elucidates the notion of an a priori concept through consideration of causalit..
Kant invites us to consider the proposition that everything which happens has a cause. In witnessing some event, Say a baseball smashing the fiont window of someone's home (causing much chagrin to the homeowner!), we might make the judgment that the bal1 caused the window to break, and common sense usually holds that we are quite correct in making that judgment. But is cause itself perceived sensibly? 1s it some apparatus which rnakes itself available to our senses for inspection? Kant would argue that the notion of a cause does not reside in the events as witnessed. Rather it is, according to him, a necessary a priori connection reason makes in linking two events in time.36
One might wonder to what sort of object might an a priori concept refer when making an a priori judgment? A posteriori concepts refer to 'sensible' perceptions, but since an a priori concept is independent of sense-experience we mutask whether an a priori concept rnakes reference to anything at dl. This is a most troubiing question since if there is nothing that an a priori concept refen to, there is then nothing about which one can make a synthetic a priori judgment. It is necessary to ask then, how a priori judgments are possible. Kant believes that our minds, and the extemal reality our minds are situated in are such that a priori perceptions are possible, and even necessary.
5. The form of perception
Kant writes that "in whatever way and by whatever means a cognition may refer to objects, the immediate way of it's refemng to them, and the aim of al1 thought as a means. is intuition."" This seems to imply that the means of reference, even for a priori knowledge, is intuition. Again, by intuition Kant means the presentation of some object to our minds pnor to any act of thought. That is to Say, an intuition of some object is it's being made readily avaiiable to my mind in a perception. The form of intuition is more like an inbuilt capability of the mind to have perceptions. The distinction between a perception and the form of that perception is an important one. That which is perceived, which is 'sensed', be it a table or chair or cornputer screen, Kant cails the matter of a perception.38The matter of perceptions may Vary, but Kant argues that there is an aspect of the act of perception itself which is invariant. This he cails the form of perception, and
"Kant, Critique, 16.
381bid. "the form must lie ready, for ail sensation, a priori in the mind."j9 In other words, we can7tsense the forrn of a perception. The form is just there sornehow, inside our minds available for presentation of the objects of our perceptions.
Now, we've seen that Kant believes that objects can be presented to the mind ernpincally, in an a postenori presentation, or independentiy of sense impressions in an a priori presentation. Kant calls the a priori presentation pure. Corresponding to a pure presentation there mut be a pure form a priori in the mind. This pure form Kant calls pure intuition, and he contends that there are really two pure forms of intuition, namely. space and time. Without situation in space and time the perception of objects is inconceivable, or so Kant argues, and so space and time are necessary for perception.
6. Space and time
Just as an a posteriori concept refers to an object sensed in perception, it may be that an a priori concept refers to some object which is independent of experience and sense perception. Kant argues that space and time are both such objects. The stmctures of these two objects are according to Kant described by the synthetic a priori judgments and concepts of geomehy and arithrneti~.~~Kant argues that the mind senses extemally and internally. The outer sense allows us to perceive objects extemal to us in space. The inner sense presents perceptions through relations in time. It therefore seems reasonable at this point to ask, as Kant himself does, just what space and time are.
391bid.
'OKorner, Kant, 29. Neither space nor tirne are abstractions fiom sense experience. In order to perceive an object extemal to myself, the concept of space must already be present.
Sirnilady objects could not be presented to my mind successively, nor simultaneously if time was not already "underlying a priori.'"' Further, both space and time are necessary to perception. While space and time cm be emptied of objects, Kant argues that it is inconceivable that there is no space, or that time can be suspended. Without space what is a ball? The very description of it requires space. Without time, what is a thrown ball? It is nothing, for without time one cannot witness the ball's spatial change. Space and time as a priori intuitions make perception possible. Al1 this, however, doesn't necessitate that space and time actually are pure intuitions, for they may simply be synthetic a priori concepts referencing something more basic to perception than themselves. Against this view Kant suggests that space and time are irreducible: there is nothing more fundamental to sense perception. Neither space nor time are concepts which cm be built from relations of things. Space, for instance, is unitary. If a ball can be perceived in its own "bal1 space", and a window in its "window space", the general space in which al1 objects reside cannot then be derived from the relation of. for instance, separateness between bal1 and window spaces. Bal1 space and window space are merely parts of the greater indivisible whole, for space itself must precede the concepts of ball space, window space and separateness.
Space underlies a priori perception. Time is similarly indivisible as different times cannot be simultaneous but must corne successively.
To sum up, space and time are pure intuitions, according to Kant, inside of
4'Kant, Critique, 20. which we "perceive". They are the objects to which synthetic a priori judgments ultirnately make reference. A priori knowledge arises through the mental inspection (in the pure intuitions) of concepts, and the synthetic a prion judgments we rnight make about those concepts can expand ouknowledge by adding to the concept what was not already present in it. By way of simple analogy, space and thecm be considered irrernovable eyeglasses through which we see and understand objects. With this basic understanding of Kant's views on intuition we can now explore a constnictivist philosophy of mathematics that has been explicitly shaped by Kant's ideas about intuition: L.E.J. Brouwer's intuitionisrn. CHAPTER 3
CERTAIN CONSTRUCTIONS?
1. Building on intuition
While Kant's pure intuition of space may be regarded by some as suspect in light of non-Euclidean geometries, his pure intuition of time has garnered considerable respect among certain philosophers of mathematics. Brouwer believes Kant's pure intuition of time is the correct mechanism through which fully consistent mathematical knowledge cm be built.'? Brouwer's intuitionism is however a radical departure fiom cIassicai mathematics. It is Brouwer's contention that classicai mathematics uses invalid reasonings in its pursuit of mathematical tmth, thus calling into question much of what classical mathematics is inclined to cal1 'tr~e'.'~In intuitionism Brouwer attempts a complete reconstruction of mathematics so that we might be assured of the tmth and certainty of at least some mathematics.
Brouwer was at variance with the manner that classical rnathematicians, and formalists in particular, approached the practice of mathematics. Brouwer believed that the formalist approach to mathematics is flawed at its tore? These flaws are well
"Brouwer, "htuitionism and Formalism" in Collected Works, 1: 127.
')Brouwer, "Volition, Knowledge, Language" in Collected Works, 1 : 444.
"Ibid., 443.
25 expressed by Arend Heyting who argues that formalist mathematics, being dependent upon formal structures is susceptible to the misunderstandings often associated with the arnbiguity found in lang~age.~'Heyting writes that "as the meaning of a word can never be fixed precisely enough to exclude every possibility of misunderstanding, we can never be mathematically sure that the forma1 system expresses correctly our mathematical thought.'* Nevertheless, Brouwer believes that mathematics can be done "properly" and if done so, can provide mathematical knowledge of which we cm be certain. This will require, as shall be noted, that mathematical activity be cleansed of the uncertainties which can arise when it is treated like an inexact language.
2. Brouwer's intuitionism in brief
As has already been noted, Brouwer owes a great debt to Kant for his conception of 'proper' mathematics, and mathematical thinking. In adopting Kant's doctrine of the pure intuition of time Brouwer believes that he can completely separate the practice of mathematics from the language used to describe that practice." Mathematics, under the banner of intuitionism will then be freed fiom the potential inconsistencies that accompany every imprecise language. This, then, requires that intuitionism be somehow
"Recall Russell's paradox.
46~endHeyting, Intuitionism: An Introduction, (Amsterdam: North Holland Publishing Company, 197 1), 4.
47Brouwer,"Histoncal Background, Principle and Methods of Intuitionism" in Collected Works, 1: 509. "language~ess.'*~For Brouwer, mathematics arises in the human mind which is predisposed, as it were, to know mathematical objects independently of sense experience in a mathematical intuition. He writes that proper intuitionist mathematics "considen the falling apart of moments of life into qualitatively different parts, to be reunited only while remaining separate by the, as the fûndamental phenomenon of the human intellect, passing by abstraction fiom its emotional content into the fhdamental phenomenon of mathematical thinking, the intuition of the bare two-on en es^.'^^ In other words, the hurnan mind is so "fashioned" that it can perceive a move of time. In so doing the mind can grasp this "move" in the form '%.hg in time and again thing."50 Then, through construction in the pure intuition of time al1 the ordinals are made known to us.
Supposing that the construction of successive numbers cm continue without limit, we ultimately reach the potentially infinite ordinal o.Brouwer outlines how that construction is possible in his doctoral thesis.
Brouwer begins his doctoral thesis with the words "One, two, three, ... we know by heart the sequence of these sounds (spoken ordinal nurnbers) as an endless row, that is to Say continuing forever according to a law, known as being fixed."" Just what this
"Brouwer, "Historical Background, Principles and Methods of Intuitionism" in Collected Works, 1510.
49~.~.~.Brouwer, "Intuition and Fomalism" in Philosophy of Mathematics: Selected Readin~s2" Ed., eds. Putnam and Benacerraf (Cambridge: Cambridge University Press, 1983), 80.
'O~rouwer,"On the Foundations of Mathematics", in Collected Works, 153 "law" is, and how we "know" it, Brouwer does not Say. He may simply mean the perception of a move of time in mathematical intuition. Nevertheless, the sequence he describes is one which unfolds in time. Brouwer then proceeds to explain how intuitionistic addition and multiplication arise. In writing dom two identical sequences, one on top of the other, (stopping at 12 for simplicity) of natural numbers as follows:
Brouwer argues that immediately there aises a one to one correspondence between the two sequences.'' If the order of the numbers of, Say, the upper sequence is changed, that one to one correspondence remains, and because of this persistence. addition and multiplication can be done. By 7 + 5, Brouwer means the following: first count up to 7, and then count on, but let the elements after 7 correspond to the sequence 1..S. The last digit reached is the sum 7 + 5." In this manner we might Say that, mathematical objects are constructed, and we know hem through metal inspection, as though our minds were able to "directly" apprehend the object. Further we now know, because of the nature of mathematical intuition, what it means to "add". The act of multiplication is similarly constnicted. Brouwer argues that the sequence of ordinal numbers can be continued "to the left" using O - 1 -2 and so on, giving rise to the integers, as well as operations on signed numbers. To get the rationals, Brouwer strangely seems to give a defnition of a rational, and goes on to Say that in the basic intuition of mathematics there is "a possibility of thinking together several entities, connected by a between, which is never exhausted by the insertion of new entities"" That is, between the ordinals one can insert5' the rationals in an ordered way. His "definition" of rationais, and how they are ordered is given as follows:
a - where b
Why Brouwer chooses to use what seems a definition of a rational nurnber instead of deriving it fiom the intuition of mathematics in way similar to his denvation of addition is unclear? Next Brouwer argues that the imtionals can be constructed fiom the previously introduced numbers (for instance as roots of algebraic equations). Thus, it seems that intuitionism is able to account for al1 the elements of classical arithmetic.
Brouwer argues that by the basal intuition of mathematics %e intuitionist recognizes only the existence of denurnerable sets"57meaning that the classical notion of the continuum is abandoned by the intuitionists. Without the theory of the continuum, however, intuitionism might seem, in Brouwer's own words, "poor and anaemic, and in particular would have no place for analy~is"~'Mathematical knowledge would be
"This problem will be discussed subsequently.
"~rouwer,"Inhiitionism and Formalism" in Collected Works, 1: 128.
58Brouwer,"Historical Background, Principles and Methods of Intuitionism" in Collected Works, 1 : 5 1 1. severely limited. He goes on to Say, though, that this fear can be alleviated through the
"second act of intuitionism which recognizes the possibility of generating new mathematical entities firstly in the form of infinitely proceeding sequences whose terms are chosen more or less freely... secondly in the form of mathematical species..."lg By species Brouwer means properties held in common by mathematical entities. Constructed mathematical entities for which the property holds are called elements of the species. This second act fùrther creates for the intuitionist the possibility of the intuitionist continuum60, and so intuitionist mathematics may not seern so empty after dl. But we mut ask whether its constructions can be considered as "certain" knowledge.
3. Constructing knowledge?
The raw materiai of intuitionist mathematics is, as Komer sumarizes it,
"intuited non-percepnial objects and constructions which are introspectively self evident.'"' The criterion of self evidence is crucial to intuitionism since objects and constructions which cannot be considered self evident are in need of something more fundamental for justification. If, after having constructed some mathematical object there still exists a need to cal1 upon an arbitrator of "self evidence", then by definition that
%3tephan Komer, The Philosoph~of Mathematics: An Introductory Essay, (New York: Dover Publications Inc., l986), 120. mathematical object, or construction, is not self evident. In order to have mathematical knowledge in the intuitionist's sense, it is thus required that we can mentally constnict and inspect mental objects. But cm we be certain of this "knowledge"? We will need to consider what it is mathematicians have at their disposal to work with; these intuited non- perceptual O bj ects.
To better understand what Brouwer rnight mean by intuited non-perceptual objects, it will be helpful to recall Kant's thought on the "matter" of mathematical a priori perception; namely synthetic a priori concepts and propositions. A triangle, for example, is not observed and then understood as its observable properties are studied. Rather, the triangle is known in the process of its being constructed. This consideration raises an important question. Can the mathematician constnict anythmg to his or her fancy, or are there restrictions on what is mathematically "knowable"? Kant writes that "in order to gain any secure knowledge a priori, [the mathematician] mut ascribe to the matter only what necessarily follows, in accordance with his concept, from his own input into it ...reason has insight only into what it produces according to its own designw6'And our reason, according to Kant, has two pure intuitions, those of space and time, meaning that synthetic a priori objects and constructions are restricted to those things which can be
"seen" and constructed in space and time. Brouwer, of course restricts mathematicians even further by accepting only the pure intuition of time as the window ont0 mathematicai tmth.
The intuitionist requires for the truth of some mathematical concept that it be
'%ant, Critiaue., 2. properly constructed. Such a view appears at first glance to be in opposition to a platonistic conception of mathematical tx-uth. Michael Dummet remarks that for the piatonist, the truth of some mathematicai statement "consists in a grasp of what it is for that statement to be true, where tnith may attach to it even when we have no means of recognizing the fact; such an understanding therefore transcends anythmg we actually leam to do when we leam the use of mathematical ~tatements.'*~In other words, the platonist in accepting the huth of those things that have not actuaily been constnicted is leaving the realm of mathematical activity, and making an appeal to some reality extemal
"which renders our [mathematical] statements as hue or false.'* By contrast, the intuitionist demands a construction as "proof '. Mathematical existence is not extenor to the mathematician, but is rather a product of the activity of the mathematician's mind.
Whenever an a priori construction has taken place an intuitionistic mathematical object exists and is justified, thus wearing "its own justification on its face." The following example will help elucidate this notion mer.
The blueprints showing the floor-plan of a house are not the house itself. One cmot move furniture into the vision of a builder who says "One day a fine bungalow will sit on this land." Only when the completed construction sits before us can we Say
%is house exists." Further, one does not need to see the blueprints used in construction to be confident that the house does indeed exist. The existence of the house finds its own justification in the fact that it has been successfblly constructed. Moreover, there are
------63MichaelDumrnet, Elements of Intuitionism., 7.
64Tbid.
32 restrictions on the house which is actually built. The house, for instance cannot be built in rnid-air, nor may it take up al1 the space on the planet. Even Bill Gates' home is fuiite in extent. The existence of mathematical entities, for the intuitionist, is similarly contingent on their construction, restricted to those things intuited for their building material, and nothing else. There is no extemal justification needed for intuitionist rnathematics. We cmsee, further, that pure existence theorems accepted by classicd mathematicians as intemally consistent have no place in intuitionist mathematics. To assert, for instance, that there exists some number with certain characteristics without any known way to constnict said number is as meaningless to the intuitionist as is the declaration of a house builder to a prospective home owner, "This is the house for you!", al1 the while pointing at a barren piece of land.
4. Construction versus laaguage
Mathematics, according to the intuitionist, is not in need of any mediation, such as the logico-linguistic arguments accepted by classical mathematicians, between it and the mathematician, nor can there exist any mediation if mathematics is to be secure. The distinction Brouwer makes between autonomous mathematics, independent of language, and a mathematics dependent on logico-linguistic arguments for its tnistworthiness is a vital one. There is a difference between the mathematicai activity itself and the description of that activity. We do need to be able to tell others of our constmctions if mathematics is to be an inter subjective discipline. Brouwer's point is that the description of mathematical constructions which one mathematician may write down on paper to convey information to another mathematician is not the mathernatics itself. He suggests that "language plays no other part than that of an efficient, but never infallible or exact, technique for memorizing mathematical constructions, and for suggesting them to others; so that mathematicai language by itself cm never create new mathematical systems.'"'
Mathematicai activity is, in Brouwer's view, autonomous, independent of sense and experience, as well as language. It is the activity of mathematics which must be free of logico-linguistic arguments. Intuitionist mathernatics, because its intuitive constmctions are independent of experience and sense, and self evidently me, can make the claim to
'certain truth' much more plausibly than a mathematics dependent on language. Yet what can it rnean to construct without language?
Retuming for a moment to Kantian thought may lead to a better understanding of Brouwer's insistence that 'proper' mathematics is languageless. In Kant's view mathematicd propositions are synthetic a priori judgments about the a prion objects space and time. Mathematical judgments thus describe the structure of space and time in
Kant's view. Komer argues that the structure of space-time cannot be fully described through passively postuiating what that structure might be. Rather, a description of the a prion objects space and time requires that a construction occur. Komer provides an illuminating example contrasting postulated and constmcted concepts, which deserves to be quoted at length:
The concept of a fifieen-dimensional sphere, which is quite self-consistent, cannot be constmcted - even though we can (and must) postdate objects for it if we are to
65Brouwer."Historicai Background, Principles and Methods of Intuitionism" in Collected Work~,1 : 5 10. make the statement that in a 'space' of at least fifieen dimensions 'there exist' at least two such spheres with no point in common. We can, however, construct and not merely postulate a three-dimensional sphere... in three dimensional space. It7s construction is made possible not merely by the self-consistency of the concept 'three-dimensional sphere', but by perceptual space being what it is. The a priori construction of a physical three-dimensional sphere mut not be conkseS with the construction of, Say, a sphere of wood or metal. Yet the possibility of the physical construction is based on the possibility of the a pnori construction - the metal sphere on the sphere in space - just as the impossibility of the physical construction of a fi fieen-dimensional sphere is based on the impossibility of the corresponding a pnori con~tniction.~~
A mathematical construction, in the intuitionist sense (without Kant's pure intuition of space), is sirnilady made possible by the structure of the pure intuition time. That which is constructed funùshes mathematical reality, not merely that which is suggested. Just as a postulated fifteen dimensional sphere goes beyond what is acceptable to the Kantian pure intuition of space, Brouwer argues that logico-linguistic postdations in classical mathematics sometimes go beyond the structure of the pure intuition of time. That is to
Say that mathematics which is dependent on Ianguage, while it may seem intemally consistent, may misrepresent what can be "true" in mathematics.
An example of how Iogico-linguistic arguments may exceed "proper" mathematics for the intuitionist is seen in the use of the logical principle of the excluded middle as a method for finding mathematical tniths. The principle of the excluded middle cmbe described in terms of some proposition P. It cmbe stated that either P is tme, or the negation of P is true; there is no middle ground. In intuitionism, however, this principle fails when we think of the proposition P as the constniction P. It is unacceptable for the intuitionist to rnake the assertion "either P is constmcted, or its negation is constmcted"
(%tephan Komer, The Philosophv of Mathematics, 28.
35 because to declare that the negation of P is constructed still requires a construction; the construction of a contradiction to P. It is not enough to siniply state "1 could not constmt
P, therefore the negation of P exists" for it may be that further effort, or new techniques will allow the construction of P.
The requirement of construction means then that not only are pure existence theorems excluded from intuitionist mathematics, but so is any mathernatics dealing with total actual infinities since an actual infinity cmot be mentally in~pected.~'But to abandon wholesale mathematics of the infinite would be, in Korner's words. "a cnppling blow to classical ma the ma tic^.'^^ One rnay wonder if the limitations of construction intuitionism imposes upon itself would be the undoing of mathematics. It shall be recalled however that Kant's conception of a priori reasoning ailows for an expansion of knowledge. That is, a priori construction allows new, unperceived and not previously thought, entities to be brought into "existence". The same appiies to intuitionist mathematics. indeed Brouwer writes that %ere is a mathematics of the potential infinite, which while avoiding the perceptually and intuitively empty notion of actual, pre-existing infinite totalities, constitutes a firm, intuitive foundation of a new analysis and opens a field of development which in several places far exceeds the fiontiers of classical
67Todo so would seem to require that, first, time be achially infinite in structure, and second that the perception of that structure i.e. the perception of an infinite amount of tirne is possible.
%ephan Korner, The Philosoph~of Mathematics, p. 124. mathematicsITfi9
To sum up, in intuitionkm what we can know is what we cm rnentaily constmct and inspect. Brouwer cdls the pure intuition of time the fundamental phenornenon of the human intellect. If Kant is right. time is indeed a necessîry precondition to perception, but at the same time we are restricted to knowledge of those things perceivable and constmctable in time. How far cari such construction take us in the general quest for certain knowledge?
69Brouwer,"Historical Background, Principles and Methods of Intuitionisrn" in Collected Works, 1: 5 1 1
37 CHAPTER FOUR
DO COMPUTERS HAVE INTUITION?
How cm it be that mathematics, being after al1 a product of human thought which is independent of experience, is so admirably appropriate to the objects of reaiity? 1s human reason, then, without experience, merely by taking thought, able to fathom the properties of real things?" Albert Einstein
1. Necessary constructions
Lntuitionism speaks of the necessity of construction to show mathematical existence. Without constmction mathematics is nothing more than a mathematician's fancihl imaginings, or so goes the intuitionist's argument. An intriguing question arises here: whether the objects of intuitionist mathematics themselves have the character of necessity. If intuitionist mathematics is to have any hope of certainty, it may be that their
'%onstmctions" must be necessary in a platonic sense. Consider, for instance, the construction of the natural nurnbers. Ln the intuitionist philosophy the naturals arise because we cmperceive a move of time and a splitting apat of a moment into two different moments. According to the intuitionist, then, the human mind is so composed
''~1be1-t Einstein, Ideas an Opinions? (New York: Crown Publishers, I982), p. 233.
38 that the naWsarise simply in pure thought. A very basic description of intuitionism
rnight be that we perceive sorne basic mathematical property, and then use that perception
to create new mathematical objects.
Mathematics, for the intuitionist, is a sort of by-product of human thought
(although oftentimes much labourious thought). But hurnan thought, unfomuiately, has not show itself to be an infallible source of certain knowledge. Since this is meit may then be then that the "knowledge" which arises in intuitionistic construction has no hope of being acceptable as "certain" in the sense that anyone could ever find themselves
"fimly penuaded" of its truth. In chapter three 1discussed the idea that intuitionism
Brouwer wants for intuitionism that it be sornehow "languageless". That being the case,
Brouwer asks the question "whether the exactness of mathematics. in the sense that mistakes and misunderstandings are excluded ...can be secured by other means?" Now,
Brouwer's use of the word "exactness" need not be identical to Rosmini's conception of certainty, but there does seem a similarity in that if some piece of knowledge is understood, and somehow known to be mistake free, then that piece of knowledge may be worthy of one's firm assent. Being that the objects of intuitionistic mathematics are intuitively known (that is our minds can directly inspect the object), and supposedly self evident, it may be reasonable to think that we might be able to ascribe certainty to intuitionist mathematical knowledge. In answer to his own question above, Brouwer says the "the languageless constructions which arise fiom the self unfolding of the basic
7'~rouwer,"Volition, Knowledge, Language" in Collected Work~,1 : 443
39 intuition, are exact and tme, by vheof their very presence in the memory
Unfortwiately, as is mentioned above, and as Brouwer himself acknowledges "the human faculty of memory which must sweythese constructions. is by its nature lirnited and liable to error? This is quite a predicament for intuitionism, for on the one hand it is in the mind that we gain knowledge, and yet because of that we may not be able to trust that knowledge.
But do the products of intuitionist mathematics acquire "existence" oniy though the persistent, 'thoughtful' efforts of the mathematician, or could it be that the
"constnictions" of the intuitionist are mathematician independent entities discovered inevitably as construction occurs? The 'certainty' of these 'discovered' mathematical objects would then rest not on the fact of their construction in mathematical intuition, but in their being discovered through mathematical intuition. Unfortunately, even if mathematical entities are mathematician independent, the problem of the fallible mind remains. Yet, From the simple fact that our minds are limited it does not follow that we mut therefore be completely uncertain of our mathematical knowledge. It might be that we "see through a glas darkly" when we mentally inspect some mathematical object and so our knowledge may be a linle muddled. But might there be anythmg about mental objects themselves, or possibly the constructive process by which we obtain the mental objects which in Godel's words, somehow make them "force themselves on us as being tr~e?"'~In other words, can mathematics itself teach anything about mathematics?
2. Construction as computation
Computation is something most of us are familia. with, at least to some extent.
Adding and subtracting numbers. for exarnple, is one of the first forms of computation to which children are introduced in school. Children are usually taught adding and
subtracting beginning with very concrete exampies such as %vo fingers and two more
fingers make four fingers". Once this sort of elementary computation is mastered. the children can then move on to less concrete examples where the numbers involved have no
"real world" significance, in the sense that adding on one's fingers does. For instance any child asked to surn, Say 2 492 and 37 986, would be il1 advised to try to use his or her
fingers. Rather, by the time t!!at this sort of mathematics is taking place the child will hopefully have been taught how to add in columns. That is, the child will have been given aprocedure by which the correct result can be obtained. Now, this concept of a procedure rnay threaten intuitionism, for what need of mental construction is there if a mechanical procedure can give the self-same results? Let me illustrate this by way of example.
Recently I wrote a very simple BASIC program and ran it on my computer. The code of the program follows:
"Kurt Godel, "What is Cantor's continuum problem?'in Philoso~hvof Mathematics, eds. Benacerraf and Putnam, 484. 30 PRINTj 40 GOTO 20
What this program essentially does is count indefinitely beginning with one (at least its output begins with one). In a sense it is mimicking what Brouwer said is the beginning of intuitionist mathematics: the unfolding of a sequence of natural numbers." It should be noted, though, that what is happening in a cornputer running this program seems to be occurring independently of any mathematician's mind. Furthemore, this simple algorithm is abundantly more effective than any hurnan mind at "unfolding" the sequence of natural numbers one by one (maybe if we had digital computers for minds we could have better reason to hope our mathematical knowledge to be "exact7'). For the sake of this argument 1 allowed the algorithm to run for thirty minutes, and in that time my computer managed to reach the number 10 083 83 1. Such a feat is impossible for the mind of even the most adept mathematician. This is not to imply that the human mind cmot count to ten million, but it would take substantially much more time.
Of course the computer could not have corne up with the numben fiom one to ten million in ten thousand lifetimes, let alone thirty minutes, had not a human mind first constructed an algorithm allowing the computer to perform the computations. On the other hand, neither could the intuitionist constnict the naturals simply on the basis of the pure intuition of a move of time. The intuitionist also needs, as we saw, the concept of one to one correspondence for intuitionistic addition to be available. Thus it seems that the 'constructability' of mathematical entities such as the natural nurnbers is dependent
------"Recall the first words of his dissertation: "One,two, three ..."
42 upon the possibility offirst being able to constmct an algorithm, or systematic procedure, by which those entities can be 'made'. The intuitionist must have the means to reason through (or to) his or her 'construction', and this cdls into question the intuitionist's claim that they can actdly perform "languageless" mathematics. Consider again the case of intuitionistic addition. To perform the sum 7 + 5 we begin by 4'observing" the unfolding of a sequence, but we must have in our minds the notion of stopping in a specific place (7), and when we begin again we must establish a one to one correspondence between the rest of the numbers in our unfolding sequence, and the subsequence 1..S. Brouwer gives no indication as to how al1 this is (or even can be) done without an intermediary language. The intuitionist might argue that he or she has available to their mind (by retention in memory) fiom previous constmctions the mental constmction we cd1 '7' when discussing (but not doing!) mathematics, and '5' also, which therefore allows the construction of' 12' to be made. With 5 and 7 available, simple mental inspection (without language) is al1 that is needed to constmct 12.1 may gant this argument, but I am left to wonder just what are these things which are supposedly lodged in the memory of the intuitionist.
Are they actuai "things" or merely "representations" of the unf'olding sequences of the mathematical intuition? It would seem that they are necessarily actual objects, for if they merely stand in reference to previously observed "unfoldings" of the mathematical intuition, one cannot help but sense language entering the picture again. If this is so, that the constructions of intuitionist mathematics are real objects residing in memory, does that mean that a cornputer which has been programmed to add "intuitionistically", and store it's results in memory (or even worse on a diskette) has actual mathematical objects residing in its memory? And what about if the "object" were stored on the diskette as a file? Has the cornputer programmer succeeded in removing what is supposedly a purely mental construct fiom the mind, storing it in a non mental form of memory? If intuitionism truly is somehow languageless, then it is might seem unlikely that the computer, or the diskette, have an actual object in "storage", as a computer language was used in the "construction" of that object. But it can be argued that a computer program is a fonnalized reasoning process. Thus it may be worth asking which is the actual mathematical construction; the 12 fiom the 7 and 5, or an algorithm (or reasoning process) which allows the construction?
'Unfolding' (that is getting the output of) an algorith., on a computer for instance, is essentially a trivial matter. It rnay be that very powerful cornputers are needed to run certain algorithms, but this is ultimately irrelevant to intuitionist mathematics.
What is important is the fact that any sort of computation can be done at ail. Why shouid we be able, for instance, to add numbers, gaining as a result a new, different number? It is the ability to reason 'computationally', if you will, which allows mental construction to occur. This is in no way meant to imply that the results of 'computation', that is the end product of doing mathematics, are insignificant. Quite to the contrary these objects are vitally important. Where would Einstein and his colleagues be without the naturals for example? The point 1am attempting to malce is that the 'output' of 'computational reasoning' may not be constructions at dl. It may be, rather, that it is the reasoning processes themselves which are the intuitionist's constructions which when utilized bring to the mathematician's mind the (possibly maîhematician independent) mathematical
objects such as the naturals. We seem to always corne back to the languageless reasoning
processes of intuitionism which can sometimes be simulated through the use of
algorithms. To see this, a closer examination of algorithms is called for.
3. Compressions and the uncornputable
Ln this age of cornputers algorithms play an indispensable role in mathematics,
and particularly in applied mathematics. The cornputer has been called a tool of
exploration for the rnathematician just as is the telescope for the astronorner. and the
microscope for the microbiologist. Whenever we have an algorithm which cm compute
some bit of information, we Say that the algorithm 'compresses' that information, and that the information itself is 'computable'. The above computer code, then, cm be called a compression of the natural numbers. Intuitionist mathematics, when viewed fiom an
'algorithmic' vantage point, might be called an intuitive construction of compressions.
The intuitionist, using her or his inbuilt ability to reason develops algorithmic processes which produce some kind of output.
The fact that aigorithms are so effective at simulating mathematical structures is more than a little remarkable. Recently, with the advent of computer graphies, the simulations being done have been astounding. Consider the following system of four affine transformations with weighted probabilities: 1 Transformation 1 Translation 1 Probability ] I
X A vertical vector ( y )can be used as coordinates which are transformed under the above set by randomly selecting one of the transformations, given its probability, and multiplying our vector by the transformation matrix, and then adding on the translation. If this process is repeated indefinitely on each resultant vector, that is if the vector is iterated through the system, and at each iteration the new point is plotted, a remarkabie shape begins to appear:
Spleenwort Fem
46 This sort of system, known as an iterated function system, has been studied by many people, including Michael Barnsley. Iterated function systems are deeply interesting mathematically, but it is their potential philosophical significance which I wish to examine here. We have a set of four rather elementary transformations which 'cornpute' something almost "lifelike". It would seem aimost as if nature itself were being simulated by the mathematics, and conversely that the mathematics cm, in some sense, cornpre~s'~ nature. Can this really be so?
In the 1930's the mathematician Alan Turing first conceptualized the basic idea of a computer. Turing was interested in attempting to find an answer to a question frst posed by David Hilbert, known as the decision problem, which essentially asked whether there is a procedure which can determine, in a finite number of steps, the tmth value of any mathematical statement. If such a procedure did exist, Turing believed that a machine could be designed which might be able to cany out that procedure, aIgorithmically so to speak. This sort of machine has been given the name "Turing Machine", and what it essentially does is convert one string of symbols into another string using a predefined, finite set of desfor that conversion. As it turned out the output of a Turing Machine could be represented as a reai number, which Turing called computable if it could be generated to unlimited accuracy using the finite set of rules. What Turing found was that there exist "uncornputable nurnbers". He did this by ingeniously using Cantor's farnous diagonaiization technique in the following way. Assuming that he was able to compile a
"1n the particular case of this "fem simulation" the compression is really quite clramatic. The above graphic is essentially compressed into only twenty eight numbers. list of every computable number, Turing considered that list written verticaily. nien by constmcting a new nurnber composed of the first digit of the first number changed by one for instance, the second digit of the second number again changed, and so on he
"cornputed" a number not on his list of al1 computable numbers. In other words, Turing found an uncomputable number. That is a number which has no "compression", or algorithm which generates it.
If we have some mathematical operation which is computable then, in John
Barrow's words. that "means it is possible to sirnulate that function by some natural fabrication or arrangement of matter and energy in the Universe."" Consider. for example that the decay of radioactive material simulates an exponential fùnction. Barrow goes on to Say that "the fact that the structure and arrangement of material and energy in the
Universe tums out to be weil described by mathematics can be ascribed to the existence of computable functions in Turing's ~ense."'~What, though, might the existence of uncomputable numbers signifi?
It is not yet clear whether aspects of the extemal world can be described in tems of uncomputable functions in an analogous rnanner to aspects of it being descnbed by computable functions. Yet the very possibility of uncomputable functions leads me to ask this question in opposition to intuitionism: 1s there mathematical truth which we cannot get to via construction? Godel's work hints at this as he found that there were true mathematical propositions whose truth could not be proven. Is construction, rather than a n~ohnBarrow, Pi in the Skv, p. 246.
''John Barrow, ibid., p. 246 'creator' of truth, merely a road to truth dready there regardless if we travel the road or not? A CERTAIN CERTAINT'Y
Now if a man believes in the existence of beautifûi things, but not of Beauty itself, and cannot follow a guide who would lead him to a knowledge of it, is he not living a drearn? Consider: does not dreaming, whether one is awake or asleep, consist in mistaking a semblance for the reality it resembles? 1 should certainly call that drearning. Contrat with him the man who holds that there is such a thing as Beauty itself and can discem that essence as well as the things that partake of its character, without ever confusing the one with the other - is he a drearner or living in a waking state? He is very much awake. So we may Say that he knows, while the other has only a belief in appearances; and might we call their states of mind knowledge and belief! ~lato~~
1. In or Out?
The search for certain tmth is by no rneans an easy one. What seems most problematic in examining the results of any such search is knowing whether we have indeed found certain truth since that knowledge seems to presuppose that we already knew what it was we were looking for in the first place. Furthemore, even knowing how to look is not without its puzzles. Does knowledge arise out of the human mind through, for instance, intuitionist construction, or does knowledge enter into the human mind as a r>Plato,The Republic of plat^, trans. F. M. Codord (London: Oxford University Press, 1943, 183. result of its being acted upon by some mind independent entities? These two positions seem at odds with one another, and would seem merto cal1 for different approaches to
"acquiring" whatever knowledge might be attainable. Yet, it may be that the two above mentioned positions are inextncably linked to one another.
Construction requires building materials. For the intuitionist we have seen that their building materials are the introspectively self evident objects intuited and constnicted in the pure intuition of time. But are the intuitionist constructions able to provide knowledge of which we can be certain? And if so, knowledge of what? If mathematics gives nothing more than descriptions, even certain descriptions, of the pure intuition of time, which is a faculty of the human mind, mathematics cannot be expected to provide any certain knowledge of what lies "beyond" the mind. Intuitionists would argue that we cm only be certain of the mathematics itself and that it would be improper to regard our mental constructions as certain representations of a "reality out there" beyond the human mind. In his book, John Barrow quotes Brouwer himself who declared that "the constmction itself is an art, its application to the real world an evil
The "certainty" that intuitionist mathematics might bring to the existence of mathematical objects should not be misrepresented as being certainty in knowledge of the real world in other words,
But is such a stance altogether warranted in view of the remarkable effectiveness of mathematics in describing real world structures? Consider for instance that in Kant's
goJohnBarrow, Pi in the Skv, p. 245. thought "natural science contains a priori synthetic judgments as prin~iples"~'meaning
that our understanding of the "red world" (which natural science investigates) might have
some certainty attached to it, if that understanding has some mathematicai bais. If then,
synthetic a priori propositions might be able to provide certain knowledge of the "real
world" (through thought aione), why do the intuitionists seem so reluctant to extend their
"certain" mathematical knowledge to knowledge of the "real world"? The intuitionist
seems to be like Plato's dreamer for while there might be some certainty in mathematical
knowledge through intuitive construction, al1 certainty is suddenly lost as soon as they
open their eyes and look out at the external world. How is it that intuitionism cmseem so
sceptical? The answer may lie in the intuitionist conception of mathematical existence.
Arend Heyting, in his Disputation, &tes that "al1 mathematicians and even
intuitionists are convinced that in some sense mathematics bear upon eternal truths, but when trying to define precisely this sense, one gets entangled in a maze of metaphysical dificulties. The only way to avoid them is to banish them from mathematics."" In the
Disputation, this statement arises from a question concerning the number of twin primes.
A twin prime is a pair of consecutive odd numbers both of which are prime, such as 3 and
5, or 1 1 and 13, whereas 13 and 1 7 are not considered twin primes. It is not known whether there are ifinitely many twin primes, or finitely many, but it is evident that the answer is either one or the other possibility. Yet, because there has been to this point no
"Kant, Critiaue, 11.
%rend Heyting, Intuitionism: An Introduction, (Amsterdam: North Holland Publishing, 1971), p. 3. construction of that answer, the intuitionist regards the question as rneaningless. Further, any mathematical construction which begins with the words "Assume that there are infinitely may twin primes ..." mut be rejected by the intuitionist since any result arising frorn such a construction cannot be considered certain. But suppose that one day it is proved that there are innnitely many twin primes. Were there not infinitely many the day before that construction? To answer this question Heyting argues that one "must refer to metaphysical concepts: to some world of mathematical things existing independentiy of ou.knowledge ...but mathematics ought not to depend upon such notions as the~e."'~
Mathematics, for the intuitionist, needs to be fiee of "rnetaphysicai concepts", but this seems a curious position since intuitionism relies so heavily on Kant's class of synthetic a prion propositions which he developed explicitly for the purpose of finding a footing for
'proper' metaphysical reasoning. How is it then, that mathematicians wanting to do proper mathematics ought not to give consideration to metaphysical concepts such as mind independent mathematical entities? There seems a double standard here.
Questions of existence where there is no construction yet, or where there may even not be any construction possible, need not be meaningless. The question whether there exist infinitely many, or finitely many twin primes is even a valid question from an analytic standpoint, for the statement "there are either finitely many, or infinitely many twin primes" can not be contradicted. And this implies that there does exist an answer to the question, even if we can not constmct that answer. A question which is (possibly) unanswerable may be analogous to Turing's uncornputable nurnben which seem to exist
--- "~rendHeyting, %id., p. 3. independently of mental construction. Thus, it may be that derthan providing mathematical objects wirh existence, construction provides the mathematician in some sense the certainty ofthe existence of mathematical objects. That is, through construction we can find answer definitively whether some mathematicai object exists, but it may be arrogant to think that through the process of construction the mathematician is actually calling into existence these entities. Simply banishing metaphysical problems fiom mathematics, as Heyting is wont to do, is unacceptable for rather than solving the existence problem it begs the question of mathematical existence. Can mind dependent thought point out mind independent entities?
2. Mathematics "out there"?
Plato was deeply interested in the comection between universal concepts. and the particulars which reflect the character of the universals. Consider, for instance, any particular chair. Chairs corne in al1 sizes and colours and styles, but they each share a common characteristic we rnight cal1 "chaimess". This cornrnon characteristic is an example of a universal concept in Plato's thought. The universal is not dependent upon the particulars, but the existence of the particulars is dependent upon the existence of the universal. Plato even asserted that there exist universals for which there may be no particular representation. Furthemore, the universal concept of chaimess exists exterior to the human mind in some 'world of universais'. This view when applied to mathematics has been called mathematical platonism, which can be said to ascribe universal mind independent existence to mathematical concepts and objects, such as number and shape.
Plato himself was very interested in mathematics believing that the not quite perfectiy straight lines and never perfectly round circles which we can construct are evidence of "ideal" straight lines and perfect circles in his world of universals. The idea of universals existing independently of the mind, though, is very puzzling. It may be, for instance, that the fact that it seems impossible to draw a perfectly straight line indicates that we cm only drearn about a line being absolutely straight. There is nothing about an
"imperfect" line which necessarily implies the existence of "perfection". Further, and more troubling, if there does indeed exist this "other world" of perfect universal concepts, how does the human mind gain access to it? If there is no means by which the mind can gain imediate access to the universals, they forever remain at best only speculation based upon perceptions of the imperfect particulars, and at worst entirely unknowable.
Where can one look for a means to this 'world of universals"? The intuitionist's basal intuition of mathematics does hold some aspect of universality to it as it is a feature of al1 hurnan minds. Still, it cannot be a universal in a platonic sense as it is by definition mind dependent. But what of the mathematical entities perceived and constmcted in the pure intuition of time? I have been discussing that they may indeed be mind independent entities, but if this is the case, the problern of their interaction with the hurnan mind still remains to be addressed. 3. Making Contact
in Kant's formulation of the pure intuition of time, he considers it to be an
"imer sense" which provides the means to perceive the activity of the mind. and it thus seems that mental intuitions and constructions have no hope of being about mind independent entities. But it does not necessarily follow that because our, sense-experience independent, perceptive apparatus is mind dependent, that it has no comection to an extemal reality.
Roger Penrose, an avowed mathematical platonist, makes the daim that "Plato's world consists not of tangible objects, but of mathematical things. This world is not accessible to us in the ordinary way but, instead, via the intellect. One's mind makes contact with Plato's world whenever it contemplates a mathematical tnith, perceiving it by the exercise of mathematical reasoning and in~ight."~Through mathematical reasoning one arrives at mathematical truth, making Penrose's description very close to that of Kant's and Brouwer's ideas conceming the perception and construction of mathematical objects in intuition. There is, however, a distinction to be made, and it is an important one.
RecalI that Kant believed that Euclidean geornetry was the a priori structure of space, but that this was later shown to be false. Nevertheless, it is empirically evident that
Euclidean geometry is a very good approximation of the structure of physical space, at least at the scale of everyday experience. And this fact, Penrose asserts, agrees well with wRoger Penrose, The Emperor's New Mind, (London: Vintage, 1WO), p. 205.
56 Plato's account of things. Remember that for Plato physical objects and structures cm
only be rough approximations of the ideal world. It is not a logical necessity, as Kant
believed for Euclidean geornetry, that this imperfect world which we inhabit merge with
the structures of the ideal world exactly. Even so, Penrose believes that meworkings of
the actual extemal world can ultimately be understood oniy in terms of precise
ma the ma tic^"'^, for the extemal world is a reflection of Plato's ideal world of
"mathematical things". Unfominately Penrose gives no indication as to how the mind
makes contact with the ideal world except to Say that it occun during mathematical
reasoning, and yet this may be the key.
4. Intuitionism and mathematicai platonism married
There are two doctrines of platonisrn, which need to be clarified. On the one hand ontological platonism asserts that the truths of mathematics are about real mathematical objectsS8&Then there is episternological platonism which "enshrines the idea that we have a means of directly perceiving mathematical object~."~'Intuitionism with it's view that intuitionistic constmctions are objects directly known by the mind might seem amenable to the doctrine of epistemological platonism. Still the intuitionist would not agree with platonism that the mathematical objects exist independently of the
"Roger Penrose, Ibid., p. 205.
"Mark Steiner, Mathematical Knowledee (Ithaca: Corne1 University Press, 1979, 109.
"Barrow, Pi in the Sky, 273.
57 mind. But in view of Turing's work which in part demonstrated the existence of numbers which are inaccessible via mental (or any other forrn of) construction, the intuitionist might reluctantly have to give some consideration to the possibility of mind independent existence for mathematical objects. As an intuitionist goes about constructing mathematical objects, his or her philosophy asserts that the mathematical objects are coming into existence, almost as though the mathematician were a creafor of mathematical knowledge. We have seen, though, that this "created knowledge seems dependent on the structure of the mentai intuition in which it is "created", which would thus appear to reduce the role of the mathematician fiom that of creator to one of a builder of knowledge, following the blueprints innate to the human mind in the pure intuition of time. Intuitionism holds, in Brouwer's own words, "that there are no non- expenenced truths"" meaning that those tniths built are not independent of mind.
However, it seems that intuitionism is in need of the existence of mind independent tmth for even the possibility of building its mind dependent û-uth. It must be true, for instance, that there exists an a priori pure intuition. We also saw that it must be true of the pure intuition that it sornehow contain the concept of one to one correspondence.
In its rnost basic assumptions intuitionism appears incapable of getting away
£Yom the existence of mind independent truth. If mathematical existence is not dependent upon construction but is rather a quality which mathematical objects hold, on their own independently of the mathematician's reasoning processes, a look at platonism does seem
"L. E. J. Brouwer, "Consciousness, philosophy and mathematics" in Philosophv of Mathematics: Selected Readines, Putnam and Benacerraf Eds., p. 90. merited. If the basic tenet of mathematicai platonism (that certain mathematical truth
exists independently of the mathematician) does seem a reasonable one to hold, the role
of the mathernatician might change again to one of an explorer searching for
mathematical truth in a strange world of ideal mathematical entities. The question of
course is, which (if any) is the best picture?
The idea that mathematical objects may be mind independent would seem to
suggest very strongly that mathematical knowledge rnight be able to be counted upon as
certain. The fallible mind problem we saw earlier is somewhat reduced in that the
existence of the mathematical object is not dependent upon oiir mind. What is needed,
then is a method of "knowing" these objects. But platonism seems lacking in a means of
actually getting at the mathematical objects, possibly rendering them beyond our mental reach. Paul Bemays, however, has suggested a duality between intuitionism and platonism which might prove a usefûl analogue for this discussion. He writes that '%he concept of number appears in arithmetic. It is of intuitive origin, but then the idea of the totality of nurnbers is superimposed. On the other hand, in geometry the platonistic idea of space is primordial, and it is against this background that the intuitionist procedures of constnicting figures takes place. This suffices to show that the two tendencies, intuitionist and platonist, are both necessq; they compliment each other, and it would be doing oneself violence to renounce one or the other."" Platonism explains why intuitionism cannot get away fiom mind independent truth, and intuitionism provides a rneans by
"~aulBemays, "On Platonism in Mathematics", in Philosoph~of Mathematics: Selected Readin~s,269. which platonism can get to that mind independent tmth.
With intuitionism and platonism brought together, the following picture for
certain truth in mathematics begins to ernerge: Certain mathematical truth exists, whether
we know that truth or not. It may even be that some of this truth is unknowable, and is
represented by those things which are incapable of being "computed" in Turing's sense.
This sort of '?ruth'' will forever have to remain uncertain to us. On the other hand, that
which we can know is capable of being computed, or in other words constructed. And
because intuitionistic constructions are irnmediately known to the mind without
intermediaries, we might have the confidence to be certain that our constructions exist, and that they are tniths about real things. There is one final consequence of the mariage between platonism and intuitionism which I would like to touch on. If as Penrose asserts the mind independent truths of mathematics are reflected in the "real world", it seems natural to hope that mathematics can redly provide a reliable picture of what is out there.
It is a big "if ', but nonetheless wonderfully stimulating to contemplate. REFERENCES
John Barrow, Pi in the Sky, (London: Penguin Books, 1993)
Paul Bemays. "On Platonism in Mathematics" Philosophy of Mathematics: Selected Readings 2" Ed., eds. Putnam and Benacerraf (Cambridge: Cambridge University Press, l983), 258-27 1.
L. E. J. Brouwer, Brouwer: Collected Works, vol. 1 ed. Arend Heyting (New York: North Holland Publishing Company, 1975)
L. E. J. Brouwer, "Consciousness, philosophy and mathematics" Philosoohy of Mathematics: Selected Readines 2"* Ed., eds. Putnarn and Benacerraf (Cambridge: Cambridge University Press, 1983). 90-96.
L. E. J. Brouwer, "Intuitionism and Formalism" Philosophy of Mathematics: Selected Readings 2206Ed., eds. Puuiam and Benacerraf (Cambridge: Cambridge University Press, 1 983), 77-90.
G. J. Chaitin. "Information, Randomness and Incompleteness" in World Scientific 2nd Edition, (1 990), 307-3 13.
G. J. Chaitin, "Randomness in Anthmetic and the Decline and Fall of Reductionism in Pure Mathematics", in EATCS Bulletin no. 50 (June 1993), 3 14-328.
Paul Davies, The Mind of God, (New York: Simon and Schuster, 1992)
Michael Dummet, Elements of Intuitionism, (Oxford: Clarendon Press, 1977)
Arend Heyting, Intuitionism: An Introduction, (Amsterdam: North Holland Publishing Company, 1971)
David Hilbert, "On the Infinite" Philosophy of Mathematics: Selected Readings 2" Ed., eds, Putnam and Benacerraf (Cambridge: Cambridge University Press, 1983), 183-201.
Imrnanuel Kant, The Cntiaue of Pure Reason, trans. Wol fgang Schwarz (Darmstadt: Scientia Verlag Aden, 1982)
Philip Kitcher, "Kant and Mathematics" in Kant's Philosoph~of Mathematics, ed. Car1 J. Posy (Boston: Kluwer Academic Publishers, 1992) Moms Kline, Mathematics and the Search for Knowled~e,(New York: Oxford University Press, 1985)
Stephan Komer, The Philosophy of Mathematics: An Introductory Essay, (New York: Dover Publications Inc., 1986)
Stephan Komer, Kant, (Baltimore : Penguin Books hc., 1955)
Roger Penrose, The Ernperor's New Mind, (London: Vintage, 1990)
Plato, The Republic of Plato, F. M. Cordord trans. (London: Oxford University Press, 1945)
Antonio Rosmini, Certain?, trans. Denis Cleary and Terence Watson (Durham: Rosmini House, 199 1)
Bertrand Russell, A History of Western Philosophy, (New York: Simon and Schuster, 1945), 7 1 2-7 18.
Mark Steiner, Mathematical Knowledee (Ithaca: Corne1 University Press, 1975) IMAGE EVALUATION TEST TARGET (QA-3)
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