FOM: geometrical reasoning, logic and proof
Detlefsen.1 at nd.edu
Wed Feb 24 13:35:07 EST 1999
Herewith a few remarks re. the recent thread of discussion regarding
geometrical reasoning, diagrams, etc.. I have enjoyed reading the elements
of this thread (yes, Robert, I am out here) and think I may have something
worthwhile to add. The thread as I see it has raised two main questions:
(i) What is the nature of diagrammatic reasoning (and what is the role of
such reasoning in mathematics)?, and (ii) What is the rightful place of
logical reasoning in mathematical reasoning ... and, more specifically, is
all mathematical reasoning rightly seen as being 'reducible' (in some
significant sense) to logical reasoning? I'll address only the second of
these two questions in this note.
Robert Black is right in saying that I have argued for a negative answer to
question (ii). More exactly, since I don't purport to know the answer to
question (ii), I have given reasons for thinking that the correct answer to
it might be negative. That's the most I can say at present. (Interested
readers can check out either of my papers 'Poincare against the logicians',
Synthese 90 (1992): 349-78 or 'Poincaré vs. Russell on the Role of Logic in
Mathematics', Philosophia Mathematica 1 (1993), pp. 1-25 for the basic
arguments.) I'll try briefly to indicate one of these reasons here ...
making special reference to Poincare, whose tersely and compressedly
expressed remarks on the subject I am trying to develop a literal content
Poincaré's views rested on what he took to be a datum of mathematical
"common sense". Anyone with sufficient mathematical experience, he
maintained, would clearly perceive an important difference between the
epistemic condition of one whose reasoning is based on the topic-blind
steps of logical inference and one whose reasoning is based on
topic-specific penetration of a particular mathematical subject. The
mathematician's inferences stem from and reflect a knowledge of a local
"architecture" (Poincaré's term) of a particular subject. The logicians, on
the other hand, represent a globally valid, topic-neutral form of knowledge
... and one that is thus insensitive to the differentiations of local
architecture. Using Poincaré's own analogy, the "logician" is like a
writer who is well-versed in grammar, but has no ideas.
In a manner strongly reminiscent of Kant's opening remarks to The First
Part of the Transcendental Problem of the Prolegomena, Poincaré therefore
stated the Problematik of his philosophy of mathematics as follows:
"The very possibility of the science of mathematics seems an insoluble
contradiction. If this science is deductive only in appearance, whence
does it derive that perfect rigor no one dreams of doubting? If, on the
contrary, all the propositions it enunciates can be deduced from one
another by the rules of formal logic, why is not mathematics reduced to an
immense tautology? The syllogism can teach us nothing essentially new,
and, if everything is to spring from the principle of identity, everything
should be capable of being reduced to it. Shall we then admit that the
enunciations of all those theorems which fill so many volumes are nothing
but devious ways of saying A is A?"
There are, however, some important differences between Kant and Poincaré.
Both recognized what might be called the "epistemic substantiveness" of
mathematics (i.e. the fact that it constitutes a significant and
substantial body of knowledge) as a datum for mathematical epistemology.
However, while for Kant it is the "apodeictic certainty" of mathematics
that is presented as the competitor of epistemic substantiveness, for
Poincaré, it is its "perfect rigor".
A closer reading, however, raises the possibility that Poincaré was not so
much intending to pose a dilemma (between epistemic substantiveness and
perfect rigor) for mathematical epistemology generally as he was to give a
critique of one particular form that such an epistemology had taken;
namely, that of a Leibniz-style logicism. For though what Poincaré presents
as contradictory are the claims that
(I) mathematics is perfectly rigorous,
(II) mathematical theorems are not merely logical truths or tautologies.
These claims are contradictory (even loosely speaking), however, only if
(I) is taken to imply something that it clearly does not imply; namely,
that the theorems of mathematics are all tautologies. In truth what (I)
seems to require is not that all theorems of mathematics be logical truths,
but rather that all inferences in a mathematical proof be logical
inferences. Thus, the second question which Poincaré asks in the
above-quoted remark (viz. "If all the propositions it enunciates can be
deduced from one another by the rules of formal logic, why is not
mathematics reduced to an immense tautology?"), which is supposed to have
no (ready) answer, would actually appear to have an easy one: namely,
"Because the axioms with which the deductions begin are not themselves
logical truths". It would not, therefore, appear to have the force that
Poincaré took it to have.
Is Poincaré's "insoluble contradiction" based on a simple-minded failure to
recognize this elementary point? The paragraph directly succeeding the one
quoted above suggests that this is not so. For there Poincare says:
"Without doubt, we can go back to the axioms, which are the source of all
these reasonings. If we decide that these cannot be reduced to the
principle of contradiction, if still less we see in them experimental facts
which could not partake of mathematical necessity, we have yet the resource
of classing them among synthetic a priori judgements. This is not to solve
the difficulty, but to baptize it; and even if the nature of synthetic
judgements were for us no mystery, the contradiction would not have
disappeared, it would only have moved back; syllogistic reasoning remains
incapable of adding anything to the data given it; these data reduce
themselves to a few axioms, and we should find nothing else in the
It was thus clear to Poincaré that the inference from the given fact that
mathematics is perfectly rigorous to the further assertion that
mathematical theorems are tautologies can be blocked by adopting the
position, open to non-logicists if not to logicists, that the axioms are
not tautologies. Still, he insisted, the problem he has in mind would not
be avoided by such a move. This is so, he explains, because even if
(contrary to logicism) it were granted that the axioms are not logical
truths, there is still the problem of explaining how the theorems of
mathematics could constitute a genuine extension of the axioms if the only
principles of inference used are such as have a purely logical character.
Poincaré's contention, then, is that, if only purely logical inferences are
used in a proof, knowledge of the theorem proved cannot constitute an
extension of whatever mathematical knowledge might be represented by one's
knowledge of the axioms used to prove it. Yet, despite this, he believed
that the conclusions of mathematical proofs typically do represent
epistemic extensions of their premises. Consequently, he was led to
conclude that not all the inferences belonging to a typical mathematical
proof can be of a purely logical character.
That, very briefly, is what I take Poincare's general view to have been vis
a vis our question (ii). As he put it:
"Mathematical reasoning has of itself a sort of creative virtue and
consequently differs from the syllogism."
SH, p. 32.
[N.B. Contrast this with Frege's remark in the preface to the Grundlagen
where he wrote:
"Thought is in its essentials the same everywhere: it is not true that
there are different kinds of laws of thought to suit the different kinds of
objects thought about ...
The present work will make it clear that even an inference like that from n
to n+1, which on the face of it is peculiar to mathematics, is based on the
general laws of logic, and that there is no need of special laws for
Famously, Poincare illustrated his general view by special reference to the
principle of mathematical induction, which he regarded as an example par
excellence of genuinely mathematical reasoning. He said a variety of
different things about induction. At the core of his conception of
induction, however, was a distinction he drew between laws and accidental
or isolated facts. He saw induction as a chief device in mathematics for
combining isolated or accidental facts into generalizations having the type
of 'unity' required to make them laws.
"We see successively that a theorem is true of the number 1, of the number
2, of the number 3, and so on - the law is manifest, we say, and it is so
on the same ground that every physical law is true which is based on a very
large but limited number of observations.
It cannot escape our notice that here is a striking analogy with the usual
processes of induction."
[N.B. Poincare distinguished between physical laws and the laws borne of
application of mathematical induction as follows:
"... an essential difference exists. Induction applied to the physical
sciences is always uncertain, because it is based on the belief in a
general order of the universe, and order which is external to us.
Mathematical induction - i.e., proof by recurrence - is, on the contrary,
necessarily imposed on us, because it is only the affirmation of a property
of the mind itself."
He then went on to describe how mathematical induction binds isolated facts
into unities and thus combines them into laws.
"For a construction to be useful and not mere waste of mental effort, for
it to serve as a stepping-stone to higher things, it must first of all
possess a kind of unity enabling us to see something more than the
juxtaposition of its elements. Or more accurately, there must be some
advantage in considering the construction rather than the elements
themselves. ... A construction only becomes interesting when it can be
placed side by side with other analogous constructions for forming species
of the same genus. To do this we must necessarily go back from the
particular to the general, ascending one or more steps. ... We can only
ascend by mathematical induction, for from it alone we learn something new.
Without the aid of this induction, which in certain respects differs from,
but is as fruitful as, physical induction, construction would be powerless
to create science."
"... we can form millions of different combinations, but any one of these
combinations, so long as it is isolated, is absolutely without value; often
we have taken great trouble to construct it, but it is of absolutely no
use, unless it be, perhaps to supply a subject for an exercise in secondary
schools. It will be quite different as soon as this combination takes its
place in a class of analogous combinations whose analogy we have
recognized; we shall then be no longer in the presence of a fact, but of a
These remarks are, of course, something less than self-evidently true and
clear. I'll now try briefly to explain what I think Poincare had in mind
... this is Detlefsen, though, and not Poincare. I know of nothing beyond
the above (and similar remarks) to suggest the view that I am describing.
So I'll let it be my view ... inspired by Poincare ... rather than
attributing it directly to Poincare. The argument, especially, is my own.
There is nothing like it in Poincare.
My thesis is this: There is something carried by reasoning by mathematical
induction that does not seem to be sheerly a product of its 'logical
content'. The sense in which this is true is the sense in which, then, I
want to say that mathematical reasoning is not 'reducible' to logical
reasoning. To illustrate my point, I ask you to consider the following
'infinitary' piece of reasoning:
(MI?): 1 has P.
If 1 has P, then 2 has P.
2 has P.
If 2 has P, then 3 has P.
Every positive integer has P.
Does (MI?) represent reasoning by mathematical induction? The answer, I
believe, is that it is not clear. We have to know more before we can say
whether or not the above is an example of reasoning by induction. What we
have to know, in particular, is whether there is a 'unity' or 'uniformity'
which binds the various premises of the form 'If n has P, then n+1 has P'
together, or whether they are just so many isolated facts which do not
share any single underlying reason in common. If there is, then I think it
is reasoning by induction. If there isn't, then I think it is not reasoning
by induction. What induction does, on this view, is to unite all instances
of 'If n has P, then n+1 has P' under a single 'reason' or 'justificatory
idea'. That idea achieves its unity by making no use of information
regarding the particular value of n. The reason why one instance of 'If n
has P, then n+1 has P' is true is (essentially) the same reason as the
reason why any other instance of it is true.
On the non-inductive reading, on the other hand, there is no single
'justificatory idea' or 'reason' why the various instances of 'If n has P,
then n+1 has P' are true. The various instances (or at least infinitely
many of them) have different 'reasons' for their truth.
Nothing changes if we modify (MI?) so as to be a logical rather than a
non-logical inference. To see this, imagine that we modify (MI?) as follows:
(MI?*): 1 has P.
If 1 has P, then 2 has P.
2 has P.
If 2 has P, then 3 has P.
For every positive integer x, either x=1 or x=2 ... or x=n
or ... .
Every positive integer has P.
(MI?*) would appear to be a logically valid inference (of some type of
infinitary logical reasoning). Note, however, that it does not necessarily
have the ability to yield its conclusion as a 'law' of mathematics. Whether
it can or not depends, once again, upon whether the premises of the form
'If n has P, then n+1 has P' have a common 'reason' or 'justificatory idea'
behind them. If they do, then the conclusion can be regarded as established
as a 'law'. If not, it cannot. I conclude from this that there is an
important feature of (MI?) and/or (MI?*)--one highly significant to its
mathematical-justificatory status--that is not reducible to the logical
contents of and/or the logical relationship(s) between its premises and
conclusion. I believe, moreover, that Poincare may have had something like
this in mind in his views on mathematical induction.
If I am right, there is a significant difference between a believer whose
execution of (MI?) or (MI?*) is inductive and one whose is not. To see this
difference, consider an (infinite) mind M (i.e. a mind that can grasp
infinitistic propositions and perform infinitistic logical operations, e.g.
infinite acts of &-introduction). Suppose that (i) M knows each of the
infinitely many propositions: P(1), P(2), P(3), ..., P(n), ... and that
(ii) she sees no connection between them ... in particular, she does not
see them as being true for "the same (essential) reason".
By an infinitary act of &-introduction, M can 'logically' infer
P(1)&P(2)&...&P(n)&... from the beliefs in (i). From this and the final
premise of (MI?*) she can then again 'logically' infer (x)P(x).
M's knowledge of '(x)P(x)' does not represent an act of genuine
mathematical insight or understanding--the grasp of a 'regularity' in the
infinite set of positive integers--but is rather an act of sheer logical
force. M may have a great capacity for such acts, but such acts can never
serve as an adequate epistemic substitute for the kind of mathematical
insight that is represented by reasoning by genuine mathematical induction.
Such, at any rate, is my view ...
Department of Philosophy
University of Notre Dame
Notre Dame, Indiana 46556
e-mail: Detlefsen.1 at nd.edu
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