TRUTH IN MATHEMATICS

truth in mathematicsOverview

Read this rather technical section not only to understand how very varied are matters of the intellect, but to appreciate the difficulties even mathematics faces in explaining itself. Profound questions — gaps, contradictions, ambiguities — lie beneath the most certain of procedures. Indeed, if pressed far enough, mathematics may be no more logical than poetry: just free creations of the human mind that unaccountably give order to ourselves and the natural world.

 

Introduction: Mathematics as Reality

Though mathematics might seem the clearest and most certain kind of knowledge we possess, there are problems just as serious as those in any other branch of philosophy. What is the nature of mathematics? In what sense do its propositions have meaning? {1}

Plato believed in Forms or Ideas that were eternal, capable of precise definition and independent of perception. Among such entities he included numbers and the objects of geometry — lines, points, circles — which were therefore apprehended not with the senses but with reason. "Mathematicals" — the objects mathematics deals with — were specific instances of ideal Forms. Since the true propositions of mathematics were true of the unchangeable relations between unchangeable objects, they were inevitably true, which means that mathematics discovered pre-existing truths "out there" rather than created something from our mental predispositions. And as for the objects perceived by our senses, one apple, two pears, etc. they are only poor and evanescent copies of the Forms one, two, etc., and something the philosopher need not overmuch concern himself with. Mathematics dealt with truth and ultimate reality. {2}

Aristotle disagreed. Forms were not entities remote from appearance but something which entered into objects of the world. That we can abstract oneness or circularity does not mean that these abstractions represent something remote and eternal. Mathematics was simply reasoning about idealizations. Aristotle looked closely at the structure of mathematics, distinguishing logic, principles used to demonstrate theorems, definitions (which do not suppose the defined actually exist), and hypotheses (which do suppose they actually exist). He also reflected on infinity, perceiving the difference between a potential infinity (e.g. adding one to a number ad infinitum) and a complete infinity (e.g. number of points into which a line is divisible). {3}

Leibniz brought together logic and mathematics. But whereas Aristotle used propositions of the subject- predicate form, Leibniz argued that the subject "contains" the predicate: a view that brought in infinity and God. Mathematical propositions are not true because they deal in eternal or idealized entities, but because their denial is logically impossible. They are true not only of this world, or the world of eternal Forms, but of all possible worlds. Unlike Plato, for whom constructions were adventitious aids, Leibniz saw the importance of notation, a symbolism of calculation, and so began what became very important in the twentieth century: a method of forming and arranging characters and signs to represent the relationships between mathematical thoughts. {4}

Mathematical entities for Kant were a-priori synthetic propositions, which of course provide the necessary conditions for objective experience. Time and space were matrices, the containers holding the changing material of perception. Mathematics was the description of space and time. If restricted to thought, mathematical concepts required only self-consistency, but the construction of such concepts involves space having a certain structure, which in Kant's day was described by Euclidean geometry. As for applied mathematics — the distinction between the abstract "two" and "two pears" — this is construction plus empirical matter. {5}

Kant, in his analysis of infinity, accepted Aristotle's distinction between potential and complete infinity, but did not think the latter was logically impossible. Complete infinity was an idea of reason, internally consistent, though of course never encountered in our world of sense perceptions. How consistent? Every schoolboy knows that infinity is something to which special rules apply. You cannot use simple mathematics to argue: infinity + 1 = infinity, so that (subtracting infinity both sides) 1 = 0. But what actually is infinity — something actual or potential? It matters very much. Some schools of mathematics avoid actual infinity because of the contradictions or antinomies that arise. Others are reluctant to do so as it bars them from many powerful and fascinating domains, from what Hilbert called "the paradise which Cantor has created for us." {6} Of course that paradise is somewhat counter-intuitive. There are hierarchies of infinite sets, infinite ordinal numbers, infinite cardinal numbers, etc. And mathematicians will take up different attitudes to such notions. A finitist like Aristotle would have accepted the existence of growing or potential infinities, but not complete ones, which would lack content and intelligibility. Transfinitists like Cantor, however, ascribed intelligibility and content even to complete infinities. And methodical transfinitists like Hilbert admitted transfinite concepts into mathematical theories because they were useful in simplifying and unifying theories, but did not believe the concepts fully existed. {7}

Mathematics as Logic

Gottlob Frege (1848-1925), Bertrand Russell (1872-1970) and their followers developed Leibniz's idea that mathematics was something logically undeniable. Frege used general laws of logic plus definitions, formulating a symbolic notation for the reasoning required. Inevitably, through the long chains of reasoning, these symbols became less intuitively obvious, the transition being mediated by definitions. What were these definitions? Russell saw them as notational conveniences, mere steps in the argument. Frege saw them as implying something worthy of careful thought, often presenting key mathematical concepts from new angles. If in Russell's case the definitions had no objective existence, in Frege's case the matter was not so clear: the definitions were logical objects which claim an existence equal to other mathematical entities. Nonetheless, Russell carried on, resolving and side-stepping many logical paradoxes, to create with Whitehead the monumental system of description and notation of the Principia Mathematica (1910-13). {8}

Many were impressed but not won over. If natural numbers were defined through classes — one of the system's more notable achievements — weren't these classes in turn defined through similarities, which left open how the similarities were themselves defined if the argument was not to be merely circular? The logical concept of number had also to be defined through the non-logical hypothesis of infinity, every natural number n requiring a unique successor n+1. And since such a requirement hardly applies to the real world, the concept of natural numbers differs in its two incarnations, in pure and applied mathematics. Does this matter? Yes indeed, as number is not continuous in atomic atomic processes, a fact acknowledged in the term quantum mechanics. Worse still, the Principia incorporated almost all of Cantor's transfinite mathematics, which gave rise to contradictions when matching class and subclass, difficulties which undermined the completeness with which numbers may be defined. {9}

Logic in geometry may be developed in two ways. The first is to use one-to-one correspondences. Geometric entities — lines, points, circle, etc. — are matched with numbers or sets of numbers, and geometric relationships are matched with relationships between numbers. The second is to avoid numbers altogether and define geometric entities partially but directly by their relationships to other geometric entities. Such definitions are logically disconnected from perceptual statements, so that the dichotomy between pure and applied mathematics continues, somewhat paralleling Plato's distinction between pure Forms and their earthly copies. Alternative self-consistent geometries can be developed, therefore, and one cannot say beforehand whether actuality (say the wider spaces of the cosmos) is or is not Euclidean. Moreover, the shortcomings of the logistic procedures remain, in geometry and in number theory.{10}

Mathematics as Exposition

Even Russell saw the difficulty with set theory. We can distinguish sets that belong to themselves from sets that do not. But what happens when we consider the set of all sets that do not belong to themselves? Mathematics had been shaken to its core in the nineteenth century by the realization that the infallible mathematical intuition that underlay geometry was not infallible at all. There were space-filling curves. There were continuous curves that could be nowhere differentiated. There were geometries other than Euclid's that gave perfectly intelligible results. Now there was the logical paradox of a set both belonging and not belonging to itself. Ad-hoc solutions could be found, but something more substantial was wanted. David Hilbert (1862-1943) and his school tried to reach the same ends as Russell, but abandoned some of the larger claims of mathematics. Mathematics was simply the manipulation of symbols according to specified rules. The focus of interest was the entities themselves and the rules governing their manipulation, not the references they might or might not have to logic or to the physical world.

In fact Hilbert was not giving up Cantor's world of transfinite mathematics, but accommodating it to a mathematics concerned with concrete objects. Just as Kant had employed reason to categories beyond sense perceptions — moral freedoms and religious faith — so Hilbert applied the real notions of finite mathematics to the ideal notions of transfinite mathematics.

And the programme fared very well at first. It employed finite methods — i.e. concepts which could be insubstantiated in perception, statements in which the statements are correctly applied, and inferences from these statements to other statements. Most clearly this was seen in classical arithmetic. Transfinite mathematics, which is used in projective geometry and algebra, for example, gives rise to contradictions, which makes it all the more important to see arithmetic as fundamental. But of course non-elementary arithmetic is not straightforward, and a formalism had to be developed. H.B. Curry was stricter and clearer than Hilbert is this regard, and used (a) terms {tokens (lists of objects), operations (modes of combination) and rules of formation} (b) elementary propositions (lists of predicates and arguments), and (c) elementary theorems {axioms (propositions true unconditionally) and rules of procedure}. But Volume I of Hilbert and Bernays's classic work had been published, and II was being prepared when, in 1931, Gödel's second incompleteness theorem brought the programme to an end. Gödel showed, fairly simply and quite conclusively, that such formalisms could not formalize arithmetic completely.

What does this mean? Suppose we postulate an arithmetical expression called X. Traditional mathematics would prove X to be either true or false. If different mathematical routes taken within the system proved that X was both true and false, however, then the system was inconsistent. If X could neither be proved as true or false within the system — and the emphasis is crucial, as the consistency could be proved in other ways — then the system is incomplete. Gödel showed that there would always be propositions that were true, but which could not be deduced from the axioms.

But perhaps even before Gödel, there were difficulties papered over. The relationship between empirically-evident statements of pure mathematics and the empirically-not evident statements of applied mathematics was unclear. Actual infinite sets were not used, but their symbols did appear in metamathematics, these being likened somewhat implausibly to stroke expressions. And then there was the question of the correctness of constructions, which involved an outlawed logic, if only minimally.{11}

Mathematics as Intuition

For intuitionists like L.E.J. Brouwer (1881-1966) the subject matter of mathematics is intuited non-perceptual objects and constructions, these being introspectively self-evident. Indeed, mathematics begins with a languageless activity of the mind which moves on from one thing to another but keeps a memory of the first as the empty form of a common substratum of all such moves. Subsequently, such constructions have to be communicated so that they can be repeated — i.e. clearly, succinctly and honestly, as there is always the danger of mathematical language outrunning its content.

How does this work in practice? Intuitionist mathematics employs a special notation, and makes more restricted use of the law of the excluded middle (that something cannot be p' and not-p' at the same time). A postulate, for example, that the irrational number pi has an infinite number of unbroken sequences of a hundred zeros in its full expression would be conjectured as undecidable rather than true or false. But the logic is very different, particularly with regard to negation, the logic being a formulation of the principles employed in the specific mathematical construction rather than applied generally. But what of the individual, self-evident experiences which raise Wittgenstein problems of private languages? Do, moreover, we have to construct and then derive a contradiction for a proposition like a square circle cannot exist rather than conceive the impossibility of one existing? And wouldn't consistency be more easily tested by developing constructions further rather than waiting for self-evidence to appear? {12}

Mathematics as Free Expression

Social constructivists took a very different line. {13} Mathematics is simply what mathematicians do. Mathematics arises out of its practice, and must ultimately be a free creation of the human mind, not an exercise in logic or a discovery of preexisting fundamentals. True, mathematics does tell us something about the physical world, but it is a physical world sensed and understood by human beings, as Kant pointed out long ago. Perhaps, somewhere in the universe, evolution has made very different creatures, when their mathematics will not resemble ours at all: it is surely possible.

Morris Kline {14} remarked that relativity reminds us that nature presents herself as an organic whole, with space, matter and time commingled. Humans have in the past analyzed nature, selected certain properties as the most important, forgotten that they were abstracted aspects of a whole, and regarded them thereafter as distinct entities. They were then surprised to find that they must reunite these supposed separate concepts to obtain a consistent, satisfactory synthesis of knowledge. Almost from the beginning, men have carried out algebraic reasoning independent of sense experience. Who can visualize a non-Euclidean world of four or more dimensions? Or the Shrödinger wave equations, or antimatter? Or electromagnetic radiation that moves without a supporting ether? Modern science has dispelled angels and mysticism, but it has also removed intuitive and physical content that appeals to experience. "We have seen the truth," said G.K. Chesterton, "and it makes no sense." Nonetheless, mathematics remains useful, indeed vital, and no one despairs because its conceptions do not entirely square with the world.

Mathematics as Embodied Mind

Jungian psychiatrists regarded numbers as archetypes, autonomous and self-organizing entities buried deep in the collective unconscious. Scientists and mathematicians have found that approach much too shadowy, lacking real evidence or explanatory power. But numbers as predispositions of inner body processes have reappeared in metaphor theory, this time supported by clinical study. Lakoff and Núñez analyze the mathematical metaphors behind arithmetic, symbolic logic, sets, transfinite numbers, infinitesimals, and calculus, ending with Euler's equation, where e, i and pi are shown to be arithmetizations of important concepts: recurrence, rotation, change and self-regulation. Mathematics is thus a human conceptualization operating with and limited to the brain's cognitive mechanisms. We cannot know if other (non-human) forms of mathematics exist, and mathematics is the language of science because both disciplines are mappings of source observations onto target abstractions, i.e. brain operations that employ innate and learned understandings of the world around us. Despite the variety and profundity of mathematics, the metaphors involved are surprising simple (if largely unconscious): object collection, object construction, measuring stick, motion along a line, container, boundary, source-path-goal, repetition (leading to models of infinity), etc. The abstract is apprehended in the concrete by conceptual metaphors, and metaphorical blends allow us to combine two distinct cognitive structures through a fixed correspondence between them: thus angles as seen as numbers in trigonometry, etc. Mathematical closure, which requires mathematical operations on numbers to always generate numbers, introduces concepts like zero, negative numbers and Boolean sets. {15}

Conclusions

As the embodied mind theory has yet to be widely accepted, there flourish today the four interpretations of mathematics: Platonism, formalism, intuitionism and social constructivism. All have their advocates, but practising mathematicians often have mixed views. A mathematician may be fortified by the Platonist view, for example, but also regard mathematics as an communal activity, one which generates deep relationships that are sometimes applicable to the "real world", a view that brings him close to social constructivism. {16}

But most mathematicians do not fish these nebulous waters. The theoretical basis of mathematics is one aspect of the subject, but not the most interesting, nor the most important. Like their scientist colleagues, they assert simply that their discipline "works". They accept that mathematics cannot entirely know or describe itself, that it may not be a seamless activity, and that contradictions may arise from unexpected quarters. {17} Mathematics is an intellectual adventure, and it would be disappointing if its insights could be explained away in concepts or procedures we could fully circumscribe.

What is the relevance to poetry? Only that both mathematics and poetry seem partly creations and partly discoveries of something fundamental about ourselves and the world around. Elegance, fertility and depth are important qualities in both disciplines, and behind them both lurks incompleteness and unfathomable strangeness. {18}

This and other pages in the theory section have been collected into a free pdf ebook entitled 'A Background to Literary Theory'. Click here for the download page.

References

1. Morris Kline's Mathematics: The Loss of Certainty (1980). Also, S. Körner's The Philosophy of Mathematics (1960), R.J. Baum's Philosophy and Mathematics (1973), and Morris Kline's Mathematics and the Search for Knowledge (1986) .
2. Chapter 1 of Korner 1960.
3. Ibid.
4. Ibid.
5. Ibid.
6. p. 73 of Korner 1960.
7. For simple introductions to modern mathematics see: Constance Reid's Introduction to Higher Mathematics (1959), Morris Kline's Mathematics in Western Culture (1972), Douglas R. Hofstadter's Gödel, Escher, Bach: An Eternal Golden Braid (1979), Bryan H. Bunch's Mathematical Fallacies and Paradoxes (1982), Ian Stewart's Nature's Numbers: Discovering Order and Pattern in the Universe (1995) and John Allen Paulos's Beyond Numeracy: An Uncommon Dictionary of Mathematics (1991).
8. Chapter 1 of Korner 1960.
9. Ibid.
10. Ibid.
11. Chapters 4 and 5 of Korner 1960, and Baum 1973.
12. Chapters 6 and 7 of Korner 1960. Also Dorothy Edgington's Intuitionism, Mathematical and David Bostock's Intuitionist Logic, both in Ted Honderich's The Oxford Companion to Philosophy (1995).
13. pp. 274-76 in John Barrow's The Universe that Discovered Itself. 2000.
14. Kline 1986.
15. George Lakoff and Raphael Núñez's Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being. (2000).
16. Reuban Hersh's What is Mathematics, Really? (1998).
17. Bunch 1982.
18. Ineffability: The Failure of Words in Philosophy and Religion, Ben-Ami Scharfstein (State University of New York Press, 1993). Q Somewhat simplistic, but with examples taken from eastern and western traditions.

Internet Resources

1. A Mathematician's Aesthetics. S. H. Cullinane. Dec 2003. http://m759.freeservers.com/ PHaesthetics.html An extended series of quotes, with references.
2. Foundations of Mathematics. 2004. http://www.britannica.com. Encyclopædia Britannica article.
3. A Century of Controversy over the Foundations of Mathematics. Gregory Chaitin. 1999. http://www.umcs.maine.edu/~chaitin/lowell.html NNA. Entertaining lecture, dwelling mostly on metamathematics.
4. Philosophy of mathematics. Jan. 2004. http://en.wikipedia.org/wiki/ Philosophy_of_mathematics. Handy introduction to the theories.
5. Plato's Middle Period Metaphysics and Epistemology. Allan Silverman. Jun. 2003. http://plato.stanford.edu/
entries/plato-metaphysics/
. Plato on knowledge, science and logic.
6. Aristotle (384-322 BCE.) Overview. 2001. http://www.utm.edu/research/iep/a/aristotl.htm. Covers all aspects of Aristotle's philosophy.
7. Aristotle's Logic. Robin Smith. Oct. 2000. http://plato.stanford.edu/entries/aristotle-logic/. Straightforward but more detailed account, with glossary and bibliography.
8. Gottfried Wilhelm von Leibniz. 1998. John J O'Connor and Edmund F Robertson. http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Leibniz.html. The man and his mathematics: extended MacTutor article.
9. Gottfried Wilhelm Leibniz. http://www.epistemelinks.com/Main/ Philosophers.aspx?PhilCode=Leib. Extensive site listings.
10. Meaning and the Problem of Universals: A Kant-Friesian Approach. Kelley L. Ross. 1999. http://www.friesian.com/universl.htm. Detailed article touching on conceptionalism.
11. Gottlog Frege. Jan. 2004. http://en.wikipedia.org/wiki/Frege. Introduction, with in-text links.
12. Friedrich Ludwig Gottlob Frege. John J O'Connor and Edmund F Robertson. Jan. 2004. http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Frege.html. Introduction on very extensive site devoted to mathematicians.
13. Gottlob Frege (1848-1925). Kevin C. Klement. 2001. http://www.utm.edu/research/iep/f/frege.htm. Simple but thorough introduction, with extensive references.
14. Gottlob Frege. Edward N. Zalta. Jan. 2002. http://plato.stanford.edu/entries/frege/ Extended entry in the Stanford Encyclopedia of Philosophy.
15. Bertrand Russell Gallery. Paul Barrette. Jul. 2002. http://www.humanities.mcmaster.ca/~bertrand/. Very full site, with much background material.
16. Bertrand Russell. A. D. Irvine. May 2003. http://plato.stanford.edu/entries/russell/. Touches on Russell's contribution to mathematical logic.
17. Inconsistent Mathematics. Chris Mortensen. Aug. 2000. http://plato.stanford.edu/entries/ mathematics-inconsistent/. Short Stanford Encyclopedia of Philosophy entry.
18. Hilbert's Program. Richard Zach. Jul. 2003. http://plato.stanford.edu/entries/hilbert-program/. A Stanford Encyclopedia of Philosophy entry: technical.
19. The modern development of the foundations of mathematics in the light of philosophy. Kurt Gödel. 1961. http://www.marxists.org/reference/subject/philosophy/ works/at/godel.htm. Excerpt from Kurt Gödel, Collected Works 1981.
20. Kurt Gödel. Aug. 2002. http://cscs.umich.edu/~crshalizi/notebooks/ godels-theorem.html. What the theorem says and doesn't say.
21. Lectures on Intuitionism: Historical introduction and Fundamental Notions. L E J Brower. 1951. http://www.marxists.org/reference/subject/philosophy/ works/ne/brouwer.htm. Excerpt from Brouwer's Cambridge Lectures on Intuitionism 1981.
22. Constructive Mathematics. Douglas Bridges. Jun. 2003. http://plato.stanford.edu/entries/mathematics-constructive/. Technical exposition of Brouwer's approach.
23. Luitzen Egbertus Jan Brouwer. Mark van Atten. Mar. 2003. http://plato.stanford.edu/entries/brouwer/ Readable introduction to Brouwer's Intuitionism.
24. Facing Up To Realism by Robert Farrell, and reply by Martin Gardner. Aug. 1981. http://www.nybooks.com/articles/6752. NY Review of Books correspondence to earlier review of The Mathematical Experience.
25. Model Theory. http://en.wikipedia.org/wiki/Model_(logic). Short entry on the models that underlie mathematical systems.
26. Model Theory. Wilfrid Hodges. Nov. 2001. http://www.seop.leeds.ac.uk/entries/model-theory/. Stanford Encyclopedia of Philosophy entry, technical in places.
28. George Lakoff. Jan. 2004. http://en.wikipedia.org/wiki/George_Lakoff. Introduction to Lakoff and controversies raised.
29. Mathematics. http://en.wikipedia.org/wiki/Mathematics. Simple introductions to many aspects of mathematics.
30. Mathematics in the Postmodernist Era. Arthur T. White. Nov. 1996. http://secure.mackinac.org/article.asp?ID=5220. Mathematics versus literary theory.
31. The MacTutor History of Mathematics archive. John J O'Connor and Edmund F Robertson. Jan. 2004. http://www-groups.dcs.st-and.ac.uk/~history/index.html. Exceptionally useful for individual mathematicians.
32. MacTutor: Links to external pages. John J O'Connor and Edmund F Robertson. Jan. 2004. http://www-groups.dcs.st-and.ac.uk/
~history/External/external_links.html
. Many excellent maths sites listed.


      C. John Holcombe   |  About the Author    | ©     2007 2012 2013 2015.   Material can be freely used for non-commercial purposes if cited in the usual way.