8.0 Deduction and Analogy in Mathematics

   In mathematical logic, deduction is analyzed as a thing in itself, as an entity entirely
independent from other mental processes. This point of view has led to dozens of beautiful ideas:
Godel’s Incompleteness Theorem, the theory of logical types, model theory, and so on. But its
limitations are too often overlooked. Over the last century, mathematical logic has made
tremendous progress in the resolution of technical questions regarding specific deductive
systems; and it has led to several significant insights into the general properties of deductive
systems. But it has said next to nothing about the practice of deduction. There is a huge distance
between mathematical logic and the practice of logic, and mathematical logic seems to have
essentially lost interest in closing this gap.
   Let us consider, first of all, deduction in mathematics. What exactly is it that mathematicians
do? Yes, they prove theorems — that is, they deduce the consequences of certain axioms. But this
is a highly incomplete description of their activity. One might just as well describe their work as
detecting analogies between abstract structures. This process is just as universal to
mathematical practice as the deduction of consequences of axioms. The two are inseparable.
   No one proves a theorem by randomly selecting a sequence of steps. And very little theorem
proving is done by logically deducing sequences of steps. Generally, theorems are proved by
intuitively selecting steps based on analogy to proofs one has done in the past. Some of this
analogy is highly specific — e.g. proving one existence theorem for partial differential equations
by the same technique as another. And some of it is extremely generalized — what is known as
"mathematical maturity"; the ability, gleaned through years of studious analogical reasoning, to
know "how to approach" a proof. Both specific and general analogy are absolutely indispensable
to mathematical research.
    Uninteresting mathematical research often makes use of overly specific analogies — the
theorems seem too similar to things that have already been done; frequently they merely
generalize familiar results to new domains. Brilliant research, on the other hand, makes use of far
subtler analogies regarding general strategies of proof. Only the most tremendously,
idiosyncratically original pieceof work does not display numerous analogies with past work at
every juncture. Occasionally this does occur — e.g. with Galois’s work on the unsolvability in
radicals of the quintic. But it is very much the exception.
    It might be argued that whereas analogy is important to mathematics, deduction from axioms
is the defining quality of mathematics; that deduction is inherently more essential. But this does
not stand up to the evidence. Even in Galois’s work, there is obviously some evidence of
analogical reasoning, say on the level of the individual steps of his proof. Although his overall
proof strategy appears completely unrelated to what came before, the actual steps are not, taken
individually, all that different from individual steps of past proofs. Analogical reasoning is
ubiquitous, in the intricate details of even the most ingeniously original mathematical research.
    And we must not forget Cauchy, one of the great mathematicians despite his often sloppy
treatment of logical deduction. Cauchy originated a remarkable number of theorems, but many of
his proofs were intuitive arguments, not deductions of the consequences of axioms. It is not that
his proofs were explicitly more analogical than deductive — they followed consistent deductive
lines of thought. But they did not proceed by rigorously deducing the consequences of some set
of axioms; rather they appealed frequently to the intuition of the reader. And this intuition, or so
I claim, is largely analogical in nature.
It is clear that both deduction and analogy are ubiquitous in mathematics, and both are present to
highly varying degrees in the work of various mathematicians. It could be protested that
Cauchy’s proofs were not really mathematical — but then again, this judgment may be nothing
more than a reflection of the dominance of mathematical logic during the last century. Now we
say that they are not mathematical because they don’t fit into the framework of mathematical
logic, in which mathematics is defined as the step-by-step deduction of the consequences of
axioms. But they look mathematical to anyone not schooled in the dogma of mathematical logic.
    In sum: it is futile to try to separate the process of deduction of the consequences of axioms
from the process of analogy with respect to abstract structures. This is true even in mathematics,
which is the most blatantly deductive of all human endeavors. How much more true is it in
everyday thought?
Kaynak: A New Mathematical Model of Mind

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