FOM: I thought this exchange would interest this list.
martind at cs.berkeley.edu
Sat Nov 1 16:53:40 EST 1997
---------- Forwarded message ----------
Date: Sat, 01 Nov 1997 01:41:44 -0200
From: Julio Gonzalez Cabillon <jgc at adinet.com.uy>
To: MATH-HISTORY-LIST at ENTERPRISE.MAA.ORG
Cc: Reuben Hersh <rhersh at math.unm.edu>
Subject: Reviewing a review
I asked Reuben Hersh about Martin Gardner's review of WHAT IS MATHEMATICS,
REALLY? Below I append his thoughtful and kind reply.
Date: Fri, 31 Oct 1997 07:28:27 -0700 (MST)
From: Reuben Hersh <rhersh at math.unm.edu>
To: Julio Gonzalez Cabillon <jgc at adinet.com.uy>
Subject: Re: Martin Gardner book review
You might be interested that Gardner wrote a review of The Mathematical
Experience in the New York Review of Books, back in the early 80's, where
he said pretty much the same thing as in his latest.
| _______________________ Begin review of Hersh book ___________________
| From: http://www.latimes.com/HOME/NEWS/BOOKS/t000091265.html
| Brought to you by the courtesy of The LA Times ... website sponsored
| by Barnes & Noble [that ought to pay the piper, huh]
| Sunday, October 12, 1997
| Mathematics Realism and Its Discontents
| WHAT IS MATHEMATICS, REALLY? By Ruben Hersh . Oxford University Press:
| 344 pp., $35
| By MARTIN GARDNER
| A physicist at M.I.T.
| Constructed a new T.O.E. [Theory of Everything]
| He was fit to be tied
| When he found it implied
| That seven plus four equals three.
Gardner repeatedly fails to address, and perhaps to
understand, the main point of my book. Mathematics
is real. The question is, what kind of real? There
are three important realities in the world: physical,
mental (subjective), and social-cultural (language,
money, politics, war, shopping, families, etc.) Traditional
philosophy ignores the third reality, which is in fact
the reality that dominates our lives. By bringing this
third kind of reality into the story, I am able to
explain what kind of reality is math.
Gardner, like all Platonists (realists) insists that math
is real. But it's not physical or mental, so what is it?
Sometimes he thinks it's a real part of the physical world,
which would eliminate most of set theory and higher-dimensional
geometry among many other kinds of non-physical math. Then
he thinks its real because it's a formal logical system,
which would say math was first invented in the late 19th
century--not true! He is oblivious to the inconsistency of
these two stories. And like all Platonists he ignores
the radical difficulty of explaining how a realm of
pure abstraction interacts with the flesh and blood
realm of real mathematical practise. His arguments
are, "Look at this! Well, gee gosh!"
| Reviewing Reuben Hersh's "What Is Mathematics, Really?" was an
| agonizing task because I have such high respect for him as a
| mathematician and such low respect for his philosophy of mathematics.
Thanks for the compliment.
| Now retired, Hersh belongs to a very small group of modern
| mathematicians who strongly deny that mathematical objects and theorems
| have any reality apart from human minds. In his words: Mathematics is a
| "human activity, a social phenomenon, part of human culture,
| historically evolved, and intelligible only in a social context. I call
| this viewpoint 'humanist.' "
| Later he writes: "[M]athematics is like money, war, or
| religion--not physical, not mental, but social." Again: "Social historic
| is all it [mathematics] needs to be. Forget foundations, forget
| immaterial, inhuman 'reality.' "
| No one denies that mathematics is part of human culture. Everything
| people do is what people do. The statement would be utterly vacuous
| except that Hersh means much more than that. He denies that mathematics
| has any kind of reality independent of human minds. Astronomy is part of
| human culture, but stars are not. The deeper question is whether there
| is a sense in which mathematical objects can be said, like stars, to be
| independent of human minds.
"can be said" is a hedge. What are they really?
| Hersh grants that there may be aliens on other planets who do
| mathematics, but their math could be entirely different from ours. The
| "universality" of mathematics is a "myth." "If little green critters
| from Quasar X9 showed us their textbooks," Hersh thinks it doubtful that
| those books would contain the theorem that a circle's area is pi times
| the square of its radius. Mathematicians from Sirius might have no
| concept of infinity because this concept is entirely inside our skulls.
| It is as absurd, Hersh writes, to talk of extraterrestrial mathematics
| as it is to talk about extraterrestrial art or literature.
| With few exceptions, mathematicians find these remarks incredible.
Gardner gives no evidence for this claim about mathematicians. I have
given dozens of talks on this subject to hundreds of mathematicians,
and find almost universal agreement! I have written numbers of
articles and books, and received much praise, virtually no criticism,
| If there are sentient beings in Andromeda who have eyes, how can they
| look up at the stars without thinking of infinity?
There are many sentient beings here on earth (both animal and human)
who have eyes and look at the stars without thinking of infinity.
| How could they count
| stars, or pebbles, or themselves without realizing that two plus two
| equals four? How could they study a circle without discovering, if they
| had brains for it, that its area is pi times the radius squared.
Euclid already knew that there are no circles in the physical world.
We invent them, based on our conditions of life here. Who knows if
there's any Euclidean geometry on Andromeda?
| Why does mathematics, obviously the work of human minds,
He's conceding my point, and adding obviously!
| have such
| astonishing applications to the physical world, even in theories as
| remote from human experience as relativity and quantum mechanics? The
| simplest answer is that the world out there, the world not made by us,
| is not an undifferentiated fog. It contains supremely intricate and
| beautiful mathematical patterns from the structure of fields and their
| particles to the spiral shapes of galaxies. It takes enormous hubris to
| insist that these patterns have no mathematical properties until humans
| invent mathematics and apply it to the outside world.
Name-calling, nothing more.
| Consider 2 1398269minus one. Not until 1996 was this giant integer
| of 420,921 digits proved to be prime (an integer with no factors other
| than itself and one). A realist does not hesitate to say that this
| number was prime before humans were around to call it prime, and it will
| continue to be prime if human culture vanishes. It would be found prime
| by any extraterrestrial culture with sufficiently powerful computers.
| Social constructivists prefer a different language. Primality has
| no meaning apart from minds. Not until humans invented counting numbers,
| based on how units in the external world behave, was it possible for
| them to assert that all integers are either prime or composite (not
| prime). In a sense, therefore, a computer did discover that 2
| 1398269minus one is prime, even though it is a number that wasn't "real"
| until it was socially constructed.
| All this is true, of course,
again he concedes my point, adding "of course"
| but how much simpler to say it in the language of realism!
Here, as in his New York Review article 15 years ago,
he mixes up the question of what is true, what is
actually the case, with the question of "simple"
or "convenient" language. I am not telling Gardner
or anyone else what language to use. I am trying to
clarify what is actually the case with regard to the
existence of mathematical objects. Talk about linguistic
convenience is beside the point.
| No realist thinks that abstract mathematical objects and theorems
| are floating around somewhere in space.
How many have you asked?
| Theists such as physicist Paul
| Dirac and astronomer James Jeans liked to anchor mathematics in the mind
| of a transcendent Great Mathematician, but one doesn't have to believe
| in God to assume, as almost all mathematicians do, that perfect circles
| and cubes have
|a strange kind of objective reality.
Yes! How strange? Why strange?
One doesn't have to believe in God to assume it, but it doesn't hurt.
|They are more that
| just what Hersh calls part of the "shared consensus" of mathematicians.
| To his credit, Hersh admits he is a maverick engaged in a
| "subversive attack" on mainstream math.
Admits? No, proclaims! But the attack isn't on mainstream Math!
It's on quite another target--mainstream philosophy of math.
| He even provides an abundance of
| quotations from famous mathematicians--G.H. Hardy, Kurt Godel, Rene
| Thom, Roger Penrose and others--on how mathematical truths are
| discovered in much the same way that explorers discover rivers and
| mountains. He even quotes from my review, many years ago, of "The
| Mathematical Experience," of which he was a co-author with Philip J.
| Davis and Elena A. Marchisotto. I insisted then that two dinosaurs
| meeting two other dinosaurs made four of the beasts even though they
| didn't know it and no person was around to observe it.
I did more than quote him, I analyzed and refuted his argument,
by explaining the distinction between numbers as adjectives
and numbers as nouns. If Gardner were serious, he would
provide my refutation, and then give his own counter refutation,
if he had one.
| A little girl makes a paper Moebius strip and tries to cut it in
| half. To her amazement, the result is one large band. What a bizarre use
| of language
It's not a matter of use of language, it's a matter of
what math really is, how it really exists.
| to say that she experimented on a structure existing only in
| the brains and writings of topologists! The paper model is clearly
| outside the girl's mind, as Hersh would of course agree. Why insist that
| its topological properties cannot also be "out there," inherent in what
| Aristotle would have called the "form" of the paper model? If a
| Hottentot made and cut a Moebius band, he would find the same timeless
| property. And so would an alien in a distant galaxy.
| The fact that the cosmos is so exquisitely structured mathematically
|is strong evidence for a sense in which mathematical properties predate
The aspects of the cosmos studied in physics yield to mathematical
That's far from saying the cosmos is altogether mathematical.
There can be no basis for such a statement except religious faith.
But it's a familiar human tendency to think that what we don't know
must look a lot like what we do know. This is a good principle for
guiding scientific research. It's not credible as a philosophical
| Our minds create mathematical objects and
| theorems because we evolved in such a world, and the ability to create
| and do mathematics had obvious survival value.
Yes, that's the point, create!
| If mathematics is entirely a social construct, like traffic
| regulations and music, then Hersh argues that it is folly
no, not folly, just mistaken
| to speak of theorems as true in any timeless sense.
|For this reason, he places great
| importance on the uncertainty of mathematics,
No, not for this reason. The reason the uncertainty of
mathematics is so important is that for centuries the
search for certainty in both mathematics and religion has
been a major motive for Platonism, or, as Gardner
prefers to call it, realism.
| but not in the sense that
| mathematicians often make mistakes. The fact that you can blunder when
| you balance a checkbook doesn't falsify the laws of arithmetic. Hersh
| means that no proof in mathematics can be absolutely certain. That two
| plus two equals four, he writes, is "doubtable" because "its negation is
| conceivable." No proof, no matter how rigorous, or how true the premises
| of the system in which it is proved, "yields absolutely certain
| conclusions." Such proofs, he adds, are "no more objective than
| aesthetic judgments in art and music."
| I find it astonishing that a good mathematician
| would so misunderstand the nature of proof. Benjamin Peirce, the father of
| philosopher Charles Peirce, defined mathematics as "the science which
| draws necessary conclusions," a statement his son was fond of quoting.
| Only in mathematics (and formal logic) are proofs absolutely certain. To
| say that two plus two equals four is like saying there are 12 eggs in a
| dozen. Changing four to any other integer would introduce a
| contradiction that would collapse the formal system of arithmetic.
| Of course, two drops of water added to two drops make one drop, but
| that's only because the laws of arithmetic don't apply to drops. Two
| plus two is always four precisely because it is empty of empirical
But a few lines up you argued that mathematical truths
are true because of their empirical content! Aren't
you ashamed to work both sides of the street?
|It applies to cows only if you add a correspondence rule that
| each cow is to be identified with one. The Pythagorean theorem is
| timelessly true in all possible worlds because it follows with certainty
| from the symbols and rules of formal plane geometry.
Formal plane geometry was unknown before the 19th century.
The name Pythagorean shows that the theorem was known much earlier! But,
you believe that, unknown to Pythagoras, Euclid, etc., the truth of the
theorem follows only from symbols and rules invented 2,000 years later.
A likely story.
| In his worst attack on the absolute eternal validity of arithmetic,
| Hersh uses the analogy of a building with no 13th floor. If you go up
| eight floors in an elevator, then five more floors, you step out of the
| elevator on Floor 14. Hersh seems to think this makes eight plus five
| equals 14 an expression that casts doubt on the validity of arithmetic
| addition. I might just as well cast doubt on two plus two equals four by
| replacing the numeral four with the numeral five.
Gardner is garbling a passage from my section on Wittgenstein.
The point is almost the oposite of what Gardner attributes to me.
I am refuting Wittgenstein's strange contention that mathematical
rules, results, and formulas are arbitrary.
| Hersh imagines that because the concept of "number" has been
| steadily generalized over the centuries, first to negative numbers, then
| to imaginary and complex numbers, quaternions, matrices, transfinite
| numbers and so on, this somehow makes two plus two equals four
Not at all. My exposition of the growth of number systems
is (1) a case history of how mathematics evolves and (2) a
demonstration that the meaning of words like "two" is not
eternally fixed, but evolves with the evolving mathematical
| It is not debatable because it applies only to positive
| integers. "Dropping the insistence on certainty and indubitability,"
| Hersh tells us, "is like moving off the [number] line into the complex
| plane." This is baloney.
Abusive language is inappropriate in scholarly controversy.
Let me "unpack" the sentence that excites Martin Gardner.
Moving into the complex plane frees us from some rules
that we can find restrictive. In the complex plane we
find a freedom that can enable great progress.
We don't find anarchy and confusion there, but more complex
The insistence on certainty and indubitability
are reponsible, at least in part, for the acceptance of
foundationist philosophies like Platonism, formalism, intuitionism,
which belie the daily experience of mathematical work.
Giving up certainty and indubitability, we can recognize
that mathematics is part of culture and society, thereby freeing
ourselves from the confusion of the foundationist philosophies.
That is the meaning of my analogy between moving
off the real line into the complex plane and moving from
philosophies which insist on certainty to a philosophy
based on the real life of mathematics.
| Complex numbers are different entities. Their
| rules have no effect on the addition of integers. Moreover, laws
| governing the manipulation of complex numbers are just as certain as the
| laws of arithmetic.
| Within the formal system of Euclidean geometry, as made precise by
| the great German mathematician David Hilbert and others, the interior
| angles of a triangle add to 180 degrees. As Hersh reminds us, this was
| Spinoza's favorite example of an indubitable assertion. I was
| dumbfounded to come upon pages on which Hersh brands this theorem
| uncertain because in non-Euclidean geometries the angles of a triangle
| add to more or less than a straight angle 180 degrees.
| Non-Euclidean geometries have nothing to do with Euclidean
| geometry. They are entirely different formal systems. Euclidean geometry
| says nothing about whether space time is Euclidean or non-Euclidian.
Spinoza knew nothing about Hilbert. His certainty about the
theorems of Euclid (shared with his contemporaries) was
not based on any formal system. It was based, among other
things, on the belief that Euclid's 5th postulate was true.
But if the truth of the postulate is dubitable, so is the
truth of its consequence, the angle sum theorem.
| Hersh's claim of triangular uncertainty is like saying that a circle's
| radii are not necessarily equal because they are unequal on an ellipse.
| Hersh devotes two chapters to great thinkers he believes were
| "humanists" (social constructivists) in their philosophy of mathematics.
| It is a curious list. Aristotle is there because he pulled numbers and
| geometrical objects down from Plato's transcendent realm to make them
| properties of things, but to suppose he thought those forms existed only
| in human minds is to misread him completely. Euclid is also deemed a
| humanist without the slightest basis. (The person most deserving to be
| on Hersh's list of maverick anti-realists is, of course, the
| mathematician Raymond Wilder. He and his anthropologist friend Leslie
| White were leading boosters of the notion that mathematical objects have
| no reality outside human culture. Hersh calls White's essay "The Locus
| of Mathematical Reality," a "beautiful statement" of social
Yes, it is.
| John Locke is on the list because he recognized the fact that
| mathematical objects are inside our brains. But Locke also
| believed--Hersh even quotes this!--that "the knowledge we have of
| mathematical truths is not only certain but real knowledge; and not the
| bare empty vision of vain, insignificant chimeras of the brain." The
| angles, in other words, of a mental triangle add to 180 degrees. This is
| also true, Locke adds, "of a triangle wherever it really exists." A
| devout theist, Locke would have been as mystified as Aristotle by the
| notion that mathematics has no reality outside human minds.
| The inclusion of Peirce as a social constructivist is even harder
| to defend. "I am myself a Scholastic realist of a somewhat extreme
| stripe," Peirce wrote in Vol. 5 of his "Collected Papers".
I appreciate these references to Locke and Peirce.
|All of Hersh's arguments against realism, by the way, were thrashed out
|by medieval opponents of realism.
I wish Gardner had named names, or even given precise references.
If he has them.
| In Vol. 4, Peirce speaks of "the
| Platonic world of pure forms with which mathematics is always dealing."
| In Vol. 1, we find this passage:
| "If you enjoy the good fortune of talking with a number of
| mathematicians of a high order, you will find the typical pure
| mathematician is a sort of Platonist. . . . The eternal is for him a
| world, a cosmos, in which the universe of actual existence is nothing
| but an arbitrary locus. The end pure mathematics is pursuing is to
| discover the real potential world."
Thanks again for interesting quotes.
| Hersh devotes many excellent
|chapters to summarizing the history of
| mathematics, and he ends his book with crisp, expertly worded accounts
| of famous mathematical proofs.
| An odd thing about this final chapter,
| though, is that Hersh writes as if he were a realist. This is hardly
| surprising, because the language of realism is by far the simplest,
| least confusing way to talk about mathematics.
Decades ago it was popular in philosophy of science to
talk (sic) as if all philosophical questions were questions
of language. As far as I know Gardner may be the last
remnant of that school.
| Over and over again Hersh speaks of "discovering" mathematical
| objects that "exist." For example, the square root of two doesn't exist
| as a rational fraction, but it does exist as an irrational number that
| measures the length of a diagonal of a square on one side.
| Mathematicians "find" complex numbers "already there" on the complex
| plane. After saying that Sir William Rowan Hamilton "found" quaternions
| while he was crossing a bridge, Hersh reminds us that quaternions did
| not exist until Hamilton "discovered them." Of course, he means that
| until Hamilton "constructed them" on the basis of a social consensus of
| ideas, they didn't exit, but his wording shows how easily he lapses into
| the language of realism.
| We must constantly keep in mind that although Hersh talks like a
| realist, his words have different meaning than they have for a realist.
| Once humans have invented a formal system like plane geometry or
| topology, the system can imply theorems that had previously been
| difficult to "discover." Their discovery is, therefore, of theorems that
| can be said to "exist" outside any individual mind, but have no reality
| beyond the collective minds of mathematicians.
| Hersh closes this chapter with a beautiful new proof by George
| Boolos of Godel's famous theorem that formal systems of sufficient
| complexity contain true statements that can't be shown true within the
| system. Boolos' proof is flawless, a splendid example of mathematical
| certainty, as are all the other proofs in this admirable chapter.
| Let the great British mathematician G.H. Hardy have the final say:
| "I believe that mathematical reality lies outside us, that our function
| is to discover or observe it, and that the theorems which we prove, and
| which we describe grandiloquently as our 'creations,' are simply our
| notes of our observations. This view has been held, in one form or
| another, by many philosophers of high reputation from Plato onward, and
| I shall use the language which is natural to a man who holds it. A
| reader who does not like the philosophy can alter the language: it will
| make very little difference to my conclusions."
| _____________________ End of review of Hersh book _________________
Gardner's book, "The Why's of a Philosophical Scrivener,"
contains chapters arguing for belief in God and for the
practise of prayer. So I understand that he might be
offended to be told that mathematics is not Divine but human.
I had no desire to offend anyone's religious sensibilities.
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