[FOM] On Foundational Thinking 1
friedman at math.ohio-state.edu
Tue Jan 20 01:17:00 EST 2004
Foundational thinking has been an essential component in many if not most of
the greatest intellectual advances throughout history. Certainly in
Aristotle, Plato, Leibniz, Frege, Godel, Turing, Newton, Einstein, Darwin,
However, such greatest intellectual advances occur with sufficient
infrequency, that despite the essential role of foundational thinking,
foundational thinking as such has no clearly defined place in contemporary
In particular, researchers generally do not deliberately apply foundational
thinking on a systematic basis in their research, nor do they generally
highlight the use of foundational thinking, nor do they teach foundational
thinking to students, nor do they explicitly discuss the role of
foundational thinking in the evolution of their subjects.
Foundational thinking is most explicitly well developed and highlighted in
connection with research in the foundations of mathematics. Here the various
deep and surprising results, going back over a Century, with roots in
antiquity, showcase just how powerful and effective and clarifying
foundational thinking can be.
(To avoid confusion, I draw a distinction between foundations of mathematics
and mathematical logic. The latter consists of various mathematical spinoffs
from foundations of mathematics, where one deemphasizes foundational
thinking, and emphasizes mathematical adventures, including connections with
various branches of mathematics.)
I believe that the deliberate, explicit, systematic, and imaginative use of
**in ANY context of depth where there is a search for systematic knowledge
of high quality, or creation of artificial systems of great utility**
will have revolutionary effect, not only in research, but in education.
Note that I have stated ** broadly enough to go far beyond science, into
engineering, and into, e.g., economics and law (economic systems and legal
systems), as well as technology.
So what is new about this kind of wild enthusiasm for foundational thinking?
After all, great philosophers like Leibniz made similar or related
predictions, and we still find ourselves in 2004 in a deeply flawed academic
culture in which foundational thinking as such is virtually invisible; and
with unnatural, counterproductive, and wholly inappropriate sharp divisions
between "disciplines", making it all the harder for change to come about. In
fact, perhaps making it all the harder for great intellectual advances to
One thing that is new about this wild enthusiasm for foundational thinking,
over what some of the great philosophers said in the past, is that
***we now have that singularly great example of explicit systematic
foundational thinking TO LEARN FROM. That is, f.o.m.***
Great philosophers of the past such as Leibniz obviously did not have access
I think it is useful to quote Godel on Leibniz, from
K. Godel, Russell's mathematical logic, 1944, in: Collected Works, vol. II.
"[Mathematical logic] On the one hand, it is a section of mathematics ... On
the other hand, it is a science prior to all others, which contains the
ideas and principles underlying all sciences. It was in this second sense
that mathematical logic was first conceived by Leibniz in his
Characteristica universalis, of which it would have formed a central part.
But it was almost two centuries after his death before his idea of a logical
calculus really sufficient for the kind of reasoning occurring in the exact
sciences was put into effect (in some form at least, if not the one Leibniz
had in mind) by Frege and Peano.
"It seems reasonable to suspect that it is this incomplete understanding of
the foundations which is responsible for the fact that mathematical logic
has up to now remained so far behind the high expectations of Peano and
others who (in accordance with Leibniz's claims) had hoped that it would
facilitate theoretical mathematics to the same extent as the decimal system
of numbers has facilitated numerical computations. For how can one expect to
solve mathematical problems systematically by mere analysis of the concepts
occurring if our analysis so far does not even suffice to set up the axioms?
But there is no need to give up hope. Leibniz did not in his writings about
the Characteristica universalis speak of a utopian project; if we are to
believe his words he had developed this calculus of reasoning to a large
extent, but was waiting with its publication till the seed could fall on
fertile ground. He went even so far as to estimate the time which would be
necessary for his calculus to be developed by a few select scientists to
such an extent "that humanity would have a new kind of an instrument
increasing the powers of reason far more than any optical instrument has
ever aided the power of vision." The time he names is five years, and he
claims that his method is not any more difficult to learn than the
mathematics or philosophy of his time. Furthermore, he said repeatedly that,
even in the rudimentary state to which he had developed the theory himself,
it was responsible for all his mathematical discoveries; which, one should
expect, even Poincare would acknowledge as a sufficient proof of its
I believe that what Leibniz could have had in mind is properly viewed, in
modern terms, as foundational thinking, in the sense that we have come to
know it and see it applied to effectively in the foundations of mathematics.
However, it was just too difficult - even for Leibniz - to be able to put
his vision into effect, until that vision had been put into dramatic and
systematic effect in the limited realm of f.o.m.
For all I know, a careful historical reading might show us that foundational
thinking cannot be what Leibniz had in mind.
But nevertheless, I propose that foundational thinking - in the sense that
we have come to know it through its dramatic applications to the foundations
of mathematics - is the appropriate modern form of what Leibniz might have
had min mind.
Some other comments on these quotations.
1. Note that Godel draws a distinction between two aspects of mathematical
logic. Its role as a branch of mathematics, and its role as the most
fundamental of all sciences.
I made essentially the same point in my parenthesized paragraph above, but
with different terminology that I think properly reflects the explosion of
activity since he wrote the quoted passages in 1944.
In 1944, there was not the kind of chasm that exists today between
mathematical logic and foundations of mathematics - and thus a chasm between
work that is called mathematical logic, and foundations of mathematics.
2. Concerning "For how can one expect to solve mathematical problems
systematically by mere analysis of the concepts occurring if our analysis so
far does not even suffice to set up the axioms?"
See my efforts in
Note that this work is suggestive of a *foundational process* which leads to
the setting up of axioms that at least interprets the axioms Godel is
talking about - the ZFC axioms.
It is not easy to state just what this foundational process is, but one
major aspect of it seems to be a recurring theme for me, and is discussed
explicitly in the above reference: simplicity investigations.
3. Concerning "Leibniz did not in his writings about the Characteristica
universalis speak of a utopian project; if we are to believe his words he
had developed this calculus of reasoning to a large extent, but was waiting
with its publication till the seed could fall on fertile ground."
I certainly do not want to claim, e.g., that, if only we developed
foundational thinking much further and much more systematically, then we
would quickly see how to prove the Riemann Hypothesis.
Foundational thinking, however, should greatly facilitate other kinds of
great advances of great general intellectual interest.
4. Concerning "Furthermore, he said repeatedly that, even in the rudimentary
state to which he had developed the theory himself, it was responsible for
all his mathematical discoveries; which, one should expect, even Poincare
would acknowledge as a sufficient proof of its fecundity."
I have certainly said many times that foundational thinking, with certain
tools and techniques, has been responsible for the preponderance of whatever
I have been able to do in f.o.m. I will try to spell out some specific tools
and techniques in later postings.
In the next postings on foundational thinking, I want to address the
following issues, not necessarily in this order.
1. What is foundational thinking? How does it differ from other kinds of
thinking? What is its relationship with mathematics and philosophy?
2. What are some examples of foundational thinking across a variety of
3. What, in detail, has been the great successes of foundational thinking in
4. What can we learn about foundational thinking from the f.o.m. example?
5. What are the main methods and tools of foundational thinking?
6. Will foundational thinking really advance our knowledge or perfect our
7. Will foundational thinking ever attain the level of depth and
effectiveness and power in connection with subjects outside mathematics?
8. Will the experience with foundational thinking in f.o.m. usably transfer
to other contexts?
9. What is foundational exposition?
10. What will a foundational exposition of mathematics look like?
11. What will a foundational exposition of subjects like statistics,
physics, economics, computer science, law, etc. look like?
12. What impact will foundational exposition have on systematic knowledge
and the creation of artificial systems?
13. What impact will foundational exposition have on education?
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