FOM: ReplyToAnand, ReplyToMattes/McLarty

Josef Mattes mattes at
Mon Nov 10 20:41:41 EST 1997

You wrote (30.Oct.)
 "I do expect that for specialists in the
 overwhelming majority of disciplines outside mathematics, there will be
 more interest in and understanding of FOM than in/of core mathematics. "

It seems to me the list of disciplines for which core mathematics is at
least as  relevant than FOM will include:
Physics, chemistry, biology, geology, astronomy, economics, engineering,
history, any field that uses probability/statistics (sociology,
psychology, ...), etc.etc.

As a rough measure, I checked today on MathSci which part of the
mathematics-related publications in certain disciplines (by MR
classification number) are referenced under foundations or under set
theory (and published after 1989). 
MR numbers:
03 Mathematical logic and foundations
04 Set theory
68 Computer science
90 Economics, operations research, programming, games
92 Biology and other natural sciences, behavioral sciences

Number of papers since 1990:
MR=92:  5976  MR=03&92:    59  MR=04&92:   25  
MR=90: 22985  MR=03&90:   224  MR=04&90:  568
MR=68: 28201  MR=03&68:  6298  MR=04&68:  365

Even in computer science, less than  quarter of the mathematical papers
refer to FOM, not to speak of other fields. 

I made a further small check: 
Going over the general interest section ("News and Views") of Nature,
the top weekly general science journal, from May to September 1997 I found
five articles relating to core mathematics: 
  Mathematics: What you eat is how you are (4.Sept.) 
  A fractal world of cloistered waves (7.Aug.) 
  Predicting where we walk (3.July) 
  Mathematics: The restless heart of a spiral (12.June)
  Field theory: Physicists learn to tie a knot (1. May)
but none that releates to FOM. 

On Tue, 4 Nov 1997, Harvey Friedman wrote:

>I wrote the following to Mattes:
>>>I hate to be repetitive, but sometimes I find telling quotes that are much
>>>stronger than anything I am saying - from people you would least expect. I
>>>would like your reaction to the Morris Kline quote from Mathematics from
>>>Ancient to Modern Times, Chapter 51, pp. 1182:
>>>"By far the most profound activity of twentieth-century mathematics has
>>>been the research on the foundations."
>I didn't get any response. I am still waiting for a response to this from
>the several people on this e-mail list that I have thrown this at. Instead
>Mattes responded with a quote from Godel that appears in another book of
>Kline - not in his major historical treatise cited above. He asks me
>whether or not I agree with the quote from Godel. My answer is: the quote
>is not self explanatory and cannot be made sense of out of context.
>As a comment to both Mattes and McLarty - note that the work that Kline
>describes in Chapter 51 starting at page 1182 is exclusively concerning
>f.o.m. in the normal customary sense that I use the phrase - and definitely
>not in any provacatively indiscriminate manner.

If I understand you correctly, 'normal customary use' refers to the
logical/set-theoretic foundations a la FRHG. In particular one can assume
that you understand the word 'foundation' in a way that Goedel would
approve of. Now the response you ask for:

You complained that some people supposedly "seek to minimize the special
importance and status of Foundations of Mathematics" and quote Morris
Kline (see above).  
First let us make clear that of course no one says Goedel's (or your) work
is unimportant. But you seem to claim much more than just that it is
 "I do not expect you to concede that the special
 importance and status of FOM is "greater than" that of fundamental
 mathematics. You may well be much more interested in core mathematics
 than you are in FOM; . . ."
So obviously you think that the importance of FOM is greater than that of
other mathematics. 

That much said, let's see what Morris Kline wrote: 

As a start, a look at your favourite ("Mathematics from Ancient to Modern
Times, published 1972, 1238 pp.). 
Does he say that foundations of mathematics are the most important part of
mathematics? No: "... the calculus, which, next to Euclidean geometry, is
the greatest creation in all of mathematics" [p.342]. 
Does this book cover all of 20th century mathematics? No: "This book
covers the major mathematical creations and developments from ancient
times through the first few decades of the twentieth century." [p.vii]
What was important about foundational work? "the development of these
several philosophies [logicism, intuitionism, formalism] was the major
undertaking in the foundations of mathematics; its outcome was to open up
the entire question of the nature of mathematics." [p.1192] (see also

I don't see how any of this supports your views. 

Eight years after "Mathematics from Ancient to Modern Times", Kline
published "Mathematics: the loss of certainty" (366 pp.). I assume he
still remembered what he wrote earlier and took this into account in his
later work. I also see no reason why imply this should be a "minor"
work (unless  you refer to physical size). 
In chapter XV (called "The authority of nature"), Kline quotes and
discusses remarks by several people (including Quine, von Neumann, Russel
and others) and goes on 
 "Perhaps more surprising is Goedel's statement of 1950 that 
    the role of the alleged "foundations" is rather comparable to the
    function discharged, in physical theory, by explanatory hypotheses...
    The so-called logical or set-theoretical foundation for number theory
    or of any other well established mathematical theory is explanatory,
    rather than foundational, exactly as in physics where the actual
    function of axioms is to explain the phenomena described by the
    theorems of this system rather than to provide a genuine foundation
    for such theorems.
 What these leaders is acknowledging is that the attempt to establish a
 universally acceptable, logically sound body of mathematics has failed."

Again, this seems to me to be quite the opposite of what you are calling
the 'genuine foundations'.

As to your assertion that the quote from Goedel is out of context: Since
Kline does not give a reference he presumably considered it unambiguous.
Not surprisingly:  
 "The so-called logical or set-theoretical foundation . . . is
 explanatory, rather than foundational, . . ." 
This quite clearly says that Goedel thinks the so-called logical or
set-theoretical foundations are not really foundational, doesn't it?  As
you said, sometimes one finds telling quotes that are much
stronger than anything one is saying - from people you would least


Finally, you too still owe us some answers: 
You wrote (25.Oct.)
 >But the main point I wish to emphasize is that we need to maintain high
 >standards for conceptually clear foundational expositions - otherwise
 >we really don't have anything which is penetrable by the general
 >intellectual community.
and I answered (29.Oct.)
 "I doubt that the problems of the intellectual community with
 mathematics stem from a lack of clarity of the concepts in mathematics. I
 rather think they come about because a.) many of our important concepts
 are quite abstract and b.) even if they are not, we like to hide them
 behind lots of technicalities."
Any comments?

I wrote 
 >>Goedel's theorem has been quoted as an example of being of interest
 >>non-mathematicians. I just wonder, how much interest would there be if
 >>were not a statement about mathematics? Without interest in
 >>mathematics/computation, who would be interested in Goedel?
You answered (25.Oct.)
 >analysis of proofs is far from a completed topic, but what has already
 >been achieved through Gottlob Frege, Godel, and others regarding            
 >analysis of proofs is far from a completed topic, but what has already
 >been achieved through Gottlob Frege, Godel, and others regarding
 >predicate calculus and formal set theory, stands tall with the greatest
 >intellectual achievments of all time. A similar kind of intellectual
 >activity later produced the first programming languages and their
 >implementation - and this stands tall with the greatest engineering
 >achievements of all time. Such is the power of the FOM outlook and
 >style of thinking. No other way of thinking about things is even
 >remotely as powerful.
I answered in turn (29.Oct.)
 "This does not seem to answer my question. It goes without saying that I
 have no intention of denying
 the importance of Goedel's results. But it seems to me that they are
 considered important because they are statements about mathematics and
 computers, which in turn are considered important. In contrast, cosmology
 for example is important because it is about nature (if you want to
 appreciate it you just have to look at a clear night sky).
 Several contributions made the point that it connects to algorithms: How
 should the importance of Goedel's work have been explained to the  
 nonexpert before the advent of computers? By explaining
 formalization of mathematics? Or was it not foundational then?"
Any comments?

Josef Mattes

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