I. Introduction
Kant's à priori - à posteriori and analytic - synthetic distinctions inaugurated
Modern epistemology and provided the architecture for knowledge in mathematics, science
and metaphysics. (1) The product of the two distinctions
yields three kinds of knowledge: synthetic à priori, analytic à priori and synthetic à
posteriori; analytic à posteriori being impossible. For Kant propositions like;
"7+5=12," "all bodies have mass" and "every event has a
cause." were synthetic and known à priorily. (2)
Post-Kantian philosophy witnessed an attack on the possibility of synthetic à priori
knowledge such as the rejections of analysis, geometry and arithmetic as synthetic à
priori by Bolzano, Helmholtz and Frege respectively. (3)
These were motivated by a fear that Kant's conceptualism, of the mind imposing space and
time on the world, may lead to anti-realism, such as that of Husserl's bracketing the
existence of the world based on his extensions of Descartes and Kant. (4)
Nominalism and idealism are anti-realist but conceptualism and conventionalism need not
be. I extend the typology of knowledge by adding knowledge by invention.
Many fundamental propositions of mathematics, science and metaphysics hence shift from the
realm of synthetic à priori to the realm of knowledge by invention. For Poincaré
fundamental definitions of mathematics are neither à priori nor à posteriori, but conventional.
I suggest that "conventional" means "known by invention." I
will argue in this paper for this unconventional interpretation of Poincaré's conventionalism.
II. Poincaré's Conventionalism
For Kant geometry and arithmetic were synthetic à priori founded on the pure
intuitions of space and time. For Frege, arithmetic was analytic à priori, (5)
and geometry synthetic à priori. (6) For Hilbert geometry
was mostly analytic à priori because of his emphasis on deductive inference, whereas the
axioms could be synthetic à priori. (7) Helmholtz attempted
to establish geometry as synthetic à posteriori. (8) Though
Poincaré maintained a Kantianism in arithmetic as its most fundamental principle, the
principle of induction, was a synthetic à priori intuition; (9)
regarding geometry, Poincaré believed that its axioms are conventions, which are
neither true nor false; and, since axioms are more fundamental than inferences: "One
geometry is not truer than another but only more convenient." (10)
If the postulates of geometry are free inventions of the mind (11)
and beyond truth value then they cannot be known. Poincaré was not a skeptic. Invention
is creation. Just as a mother who gives birth to a child knows her child better than
anyone else, Euclid who gave birth to his postulates knew Euclidean geometry better than
anyone else. Lobachevski and Riemann who gave birth to alternative non-Euclidean
geometries knew their respective geometries better than anyone else. So invention is
linked to the most intimate kind of knowledge. The use of "knowledge" here is an
extension of the traditional "justified true belief" one to which the activity
of justification is added as a fourth condition. (12)
Sentences which are neither true nor false are meaningless. Poincaré did not believe
that geometry is founded on meaningless conventions. Rather the creative dimension of his conventions
gives meaning to geometry. Poincaré explicitly states that conventions are not
arbitrary. (13) I suggest a Poincaréan reconstruction in
which the postulates of geometry have truth value and are known by their very invention of
them. Since invention is an activity, geometric postulates are a paradigm of knowledge as
actively justified true belief.
III. Knowledge by Invention
I suggest extending the à priori - à posteriori dichotomy to a trichotomy of à
priori knowledge, à posteriori knowledge and knowledge by invention. If S invents p; then
S knows p if she believes p; can justify p; p is true; and S performs the activity of
justifying p, which is the inventing itself. If I utter spontaneously "there are at
least two points in a plane which cannot be joined by a straight line," I say
something unusual which could generate a radically different geometry. However, this may
be arbitrary and not an authentic invention. A whim is not an invention and hence not
worthy of truth value.
What are the individuating criteria for inventions? Poincaré gives only one hint, that
inventions are not arbitrary. Inventions are either created or not created like Russell's
facts, which either obtain or do not obtain. (14) Even
though we cannot provide an exhaustive set of individuating conditions, the importance of
inventions in the epistemology of mathematics can no longer be marginalized. Any serious
working mathematician can relate experiences where some idea comes in a flash. Whereas an
inexperienced mathematician may fail to use this flash, a seasoned mathematician like
Poincaré or Riemann capitalizes on such a moment and invents a fundamental
generalization.
I conjecture now some conditions for inventions:
Background conditions: (a) Invention does not take place in a vacuum.
There is a background complex of the history of the discipline, for example, the history
of geometry from Pythagoras to Lobachevski in Riemann's case. (b) The inventor has
carefully been working on some problems in her discipline, which she has been unable to
resolve with the tools available to her. (c) The inventor's work is sincere and objective
insuring that she has considered all the existing alternatives available to her.
Content conditions: The invented proposition is (a) not arbitrary, (b)
chosen freely by the inventor and (c) meaningful, taking on a truth value.
Linguistic conditions: An invented proposition is (a) private in its
invention and (b) public once invented. Just as Galileo could use the telescope after its
invention by Digges, similarly Euclid's postulates, once invented, could be used by
generations of geometricians.
One consequence of these conditions combined with the definition of knowledge as
actively justified true belief is that the inventor has a greater degree of knowledge of
the invented proposition than anyone else. I believe that there are degrees of knowledge.
We always work towards ultimate knowledge which perhaps can never be attained. However, we
can know something "to the best of our knowledge." (15)
Fermat knew his last theorem better than anyone else because he invented it.
There is a glaring deficiency in our account of conventions. All conventions
could not be inventions, as this would make the use of "conventions" too
narrow. We would have to deny knowledge of the axioms of geometry to Descartes, Newton and
Gauss as they did not invent Euclid's axioms. I hence define "convention"
as:
A proposition p is conventional =def. Some person S invents p
at time t0 and S either accepts or does not accept p at time t0;
or some person S invents p at time t0 and some person T
accepts p as a convention at a later time t1.
This definition is rather cumbersome, but hopefully it captures the meaning of conventional.
I will not provide more details here as I have discussed it elsewhere. (16)
I have used "person" here to leave the definition in its most general
formulation. In most cases the "S" would refer to some great person such
as Euclid, and "T" could refer to great persons like Descartes or to
ordinary persons like me. I am also using the bold face "conventional" in
the unconventional sense and "convention" without the boldfacing in the ordinary
sense. Needless to say a complete account of conventional propositions would
involve lists of conditions for "accepting" and for "convention" which
is beyond the scope of this paper.
Are the postulates of geometry synthetic or analytic? Since for Poincaré postulates
are disguised definitions, the more general question then is: are definitions analytic or
synthetic? Traditionally, whatever is true by definition is treated as analytic, such as
in the definition of "bachelor" as "an unmarried male". Janet Folina
claims that Poincaré distinguishes between stipulative and constructive definitions. (17)
I also postulate a distinction in Poincaréan epistemology motivated by Chisolm's
distinction between a particularist and a generalist approach when searching for a
definition of knowledge. (18) A particularist presupposes
that we already have some intuitive idea of what knowledge is and look for a necessary and
sufficient criterion to crystalize our concept of knowledge and make it more precise. A
generalist claims that we must find a necessary and sufficient criterion and only then
will we have any idea of what knowledge is. (19)
Generalizing on this, we demarcate two types of definitions: The first where we already
have an intuitive idea of what the concept designated by the definiendum is and the
definition helps to make it more precise. The definitions of "point" and
"straight line" are of this type. The second type are stipulative as we already
have the concept of the definiens first and then designate the word or expression
to signify this concept. Such is the definition of "bachelor". In the second
type any other expression or word could have been used as well, while this is not true of
the former. I conjecture that the first type are synthetic while the second type are
analytic.
Are all definitions which are inventions synthetic? On the Kantian use of
"analytic" it would be odd if in an invented proposition the concept of the
predicate were to be contained in the concept of the subject, for then, an invention would
not be needed to create such a proposition. Yet, there may be some analytic propositions
in which the concept of the predicate is contained in the concept of the subject and yet
no one until the invention has seen that. Leibniz's law of the indiscernablity of
identicals and Frege's definition of number may be good examples of these. (20)
Frege distinguished between two types of analytic statements, those which have content
tell us something new about the world and those which do not. (21)
The law of identity (_x x=x) is of the former type, whereas an instance of the law of
identity such as "Socrates = Socrates" is of the latter type. I leave open the
question of whether some invented propositions may be analytic, since the outcome does not
effect the extension of the Kantian dichotomy of knowledge to the trichotomy I suggest.
Poincaré himself never claims such a trichotomy of knowledge. What I suggest then is a
reconstruction which sustains the Poincaréan spirit and his epistemological edifice more
than what he himself explicitly laid down.
My reconstruction is based on Chapter IV of Science and Hypothesis. The chapter
begins with a question of the epistemic status of geometric postulates; as to whether they
are à priori or à posteriori. Next he claims that they are not à priori; which
nonetheless does not imply that they are known through experience. Next, Poincaré shows
that they are not à posteriori. (22) Poincaré does not
deny the validity of disjunctive syllogism. Poincaré implies that the à priori - à
posteriori dichotomy is not exhaustive, hence any disjunctive syllogism constructed on it
would be an instance of the fallacy of false dilemma. What is the third possibility?
Poincaré provides the answer at the end of the chapter where he claims that the axioms of
geometry are neither à priori nor à posteriori but disguised definitions which are conventions,
free inventions of the mind which are nonetheless not arbitrary. (23)
IV. CONCLUSION
This paper has offered a suggestion which would help resolve some of the problems in
contemporary epistemology of mathematics.
Most contemporary philosophers of mathematics have matured to have expelled the
empiricism - rationalism debate from the epistemology of mathematics. Consequently, they
disguise their embarrassment with their tenuous views or non-views on the epistemic status
of axioms and definitions by merely shrugging their shoulders and claiming that it is not
their concern. This embarrassment persists because there have been no convincing arguments
to establish whether the knowledge of axioms is synthetic à priori or analytic à priori.
The dogmatic claims of logicists like Russell and intuitionists like Brouwer made the
nature of the epistemic status of axioms and even inferences an ideological debate. The
lack of any firm claims about the epistemic status of mathematical propositions does not
mean that we have evolved into a new age of the philosophy of mathematics and have
transcended the archaic synthetic - analytic and à priori - à posteriori distinctions.
This approach simply sweeps the fundamental issue in the epistemology of mathematics under
the rug.
Poincaré provides an alternative conception of the epistemological status of axioms
and definitions. I can at the moment offer no more convincing arguments for the claim that
axioms are known by invention. One lesson that we can definitely learn from Poincaré is
that definitions and axioms are among the most fundamental notions of mathematics. Any
epistemology of mathematics then must give an account of the epistemic status of these
notions. It is perhaps an enormous irony that formalism, logicism and even intuitionism;
all of which are in the spirit of Modern philosophy dominated by Cartesian
foundationalism, acknowledge that axioms and definitions are the foundations of
mathematics, hence all mathematics emerges from them; yet, they seem less worried about
the epistemic status of these foundations than they are about the status of logical
inferences used in the accountability of mathematical propositions. It is no wonder
contemporary discussions in the philosophy of mathematics have become discussions on proof
theory, on whether the inferences are mechanical or constructive. Poincaré, were he alive
today, would remind those philosophers of mathematics who are still in a foundationalist
mood that they better start worrying about them and focusing more of their attention on
the epistemic status of the foundations.
The importance of knowledge by invention for the notion of paideia is
monumental. Socrates was uncompromising in his pedagogical method where the proposition to
be known by the knower was at the end of a personal quest full of labor as made apparent
by the midwife metaphor in the Theaetetus. The role of the teacher was that of the
midwife who aids in the birthing, but it is the knower who has to give birth to the
knowledge itself. In this age of the information explosion, internet and computer
searches; there is a deceptive delusion that this quest has become easier and the
conflation of "information" with "knowledge" threatens to propel us
into some undesirable science fiction future where we become mutations of the humans that
had evolved up to our times. In any university library the hustle and bustle in the
reference room terminals in conjunction with the eerie silence in the stacks of journals
and books is an indication that the new generation has come to believe that the quest of
knowledge does not extend beyond the keyboard, whereas more than ninety percent of journal
articles are not accessible through the computer data bases, most of which are from 1980
onwards. It is ironic that more than the great philosophers Descartes, Leibniz, Locke and
even Kant; we find the most Platonist epistemology in Poincaré who was principally a
mathematician. Knowledge by invention is not only possible, but it is a paradigm of
knowledge in the Socratic-Platonic spirit. If educationists begin implementing this type
of knowledge in their pedagogical methodologies we may ultimately be able to accomplish
the revolution in education that Dewey, Montessori and Krishnamurti dreamed of.
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