Off and on for the entire history of modern mathematics, and especially
for the last century, differences concerning the most fundamental
notions about mathematics have gone unresolved. Deeply felt differences,
right at the foundations, remain.
When Fisher views the "subjective foundational views" of Russell and
Hilbert as similar, and appeals to issues of "neural structure," he's
rejecting some notions of "unquestionable truth" that many have held
dear, and questioning the rightness of the resolution of some old
conflicts in mathematics, but he has much hard won scientific knowledge
behind him. If he shows hostility in some of his passages, perhaps
that hostility is a fair reaction to superhuman pretensions.
FISHER: "Ah! So we have to be careful that we haven't described
or axiomatized our neural processes in some reprehensible way?
What sort of thing is the "real evidence" referred to here?
Something to do with consistency or coherence or non-contradiction?"
What else, indeed, do we have save an ability to check for consistency,
or coherence, or non-contradiction? The evidence that this is ALL we
have, and that the checking even at this level is very hard, has been
accumulated, through almost countless psychological experiments, over
a century.
Angels or Gods may know truths. We, as humans, know less, no matter
what we pretend.
Nor do we control or apprehend some of the best neural functions we
have. A child between the ages of 6-14 learns about 8 words per day,
day after day, with contexts and multiple definitions for each. Only
in very rare cases can the child recall when or how the words were
learned. Adults learned most of the 50,000-100,000 words they know
in this way, and continue to learn in this same unconscious fashion.
The words people know seem "intuitive." We know they are not. Many,
if not most ideas, seem to be assimilated similarly. Often these ideas
seen "intuitive" to those long familiar with them. We know they are not.
Is mathematical learning so very different? If you judge by actions,
it would seem to be the same. For generations, good math grad students
have been auditing courses, prior to taking them for credit. No doubt
this is fine academic practice. But it does cast doubt on the notion
of "intuition" applied to that coursework.
Psychology knows little enough, but it knows this much after a century
of investigation about the human mind, its capacities, and its limits.
Our minds have limits, and we are barely conscious of these limits.
As animals, we are poorly constructed to gauge certainty. Much of the
best of what we do in unconscious.
A succinct statement of the limits of human minds in the formulation
or recognition of truth is Chapter 3, "Sysreps and the Human Mind" in
CONCEPTUAL FOUNDATIONS FOR MULTIDISCIPLINARY THINKING by Stephen J.
Kline (Stanford University Press, 1995). Kline's chapter is particularly
useful for people considering human capacities for forming and checking
systems.
Another useful summary of the broadly held consensus in cognitive science
was written by E.D. Hirsch Jr. (THE SCHOOLS WE NEED: and Why We Don't
Have Them, Doubleday, N.Y. 1996, page 225.) Many useful footnotes in
that passage are deleted.
"For higher-level competencies, cognitive science has constructed
theoretical models that describe skills and learning processes.
The current consensus is that these competencies are described, at
least operationally, by schema theory- a schemata being a remembered
system of typical traits that are related to each other in typical
but adjustable ways. For example, when I read the word "ostrich"
(assuming that I understand it), I activate a schema and select
from it a range of traits and meanings that seem most appropriate
in the context. But there are other components of reading skill
that depend less on conscious-knowledge schemas and more on habitual
operations like eye movements and letter recognitions. These
constantly repeated, automated strategies (sometimes called "rules")
are learned through practice and repetition. Reading skill, for
example, includes both "rules," that is, mastery of operations
continually repeated to the point of automaticity, and schemas
consisting of interconnected frames of knowledge. In the case of
reading skill, the most important schemas are represented by
particular word meanings and cultural conventions."
"That intellectual skills should rest on rules and schemas arises
from a fundamental limitation of the mind called "short term" or
"working" memory, which is capable of retaining and manipulating
only a very limited number of items over a very brief time span
before degenerating. This fundamental limitation of thinking and
learning is physiological; it applies to novice and expert alike.
Once the very constraining limit of short-term memory is reached,
some elements disappear from awareness. The mind must strike while
the iron is hot; if it doesn't, those vanished elements will cease
to be available. For overcoming these limitations of working
memory, the mind uses rules and schemas to achieve automation and
chunking. Automation is a technique that enables the mind to engage
rapidly and unconsciously in the subroutines of an activity without
bringing them into working memory at all. Chunking is a means for
rapidly representing a whole range of knowledge elements by a single
image, word, or symbol, so that the whole array need not be present
in the mind all at once."
"In the long run, chunking, which is connected with schema
development, is more important for developing very high levels of
intellectual skill. For while the increase in competency available
from the automation of repeated operations is immensely important
in the early stages of learning, its limits may be rather quickly
reached. Once automaticity of repeated operations has been attained,
higher levels of skill depend more upon the continued acquisition,
habituation and chunking of relevant knowledge. In reading, for
example, once a plateau of rule automation has been attained, higher
levels of expertise reside almost entirely in developing, expanding,
and making accessible a reader's relevant intellectual capital."
"The distinction between rule and schema is roughly equivalent to
the traditional distinction between skill and content. . . . .
Cognitive science shows that . . . intellectual skills consist of
both form and content."
THESE HUMAN CHARACTERISTICS AND LIMITS DETERMINE HOW WE CAN REASONABLY
INTERPRET MATHEMATICAL DOCTRINE, AND THE HISTORY OF CONTROVERSIES AND
PATTERNS INVOLVING MATHEMATICAL DOCTRINE.
However wonderful our ideas and schema may be (and they are wonderful)
we aren't, as animals, sure where these ideas and schema came from.
Most of what we know and take for granted is socially constructed. For
most ideas, we have to assume that someone "told us so" or that some
individual or group "told us something about it" and we "sorted the rest
out for ourselves." If you look at math instruction, from elementary
school on, very much must be this way, and it seems impossible to
distinguish the learned from any "pure prior knowledge" that may exist.
Nor are we as animals sure when our ideas or schema have been checked,
carefully, against evidence of any kind. Human beings can't distinguish
between the "unquestionably intuitive" the "socially constructed" and the
"constructed from sensory data" parts of our conceptual lives. That must
have been true of very great human beings, such as Hilbert, Russell, and
their many followers. These facts, which must now be regarded as well
established, ought to effect our interpretations of mathematical history,
and mathematical doctrine.
If the psychological evidence is to be believed, we don't carry any
universe-based, unquestionable certainty in our heads, whether we feel
we do or not.
In mathematical history, awareness of human limits would argue for an
effort to try to discount human feelings of certainty based on subjective
notions, backed by emotion.
Suppose in mathematical arguments, and arguments about mathematical
history, "axioms" were called "assumptions" or "high status" assumptions?
Suppose mathematics were thought of as "the study of systems based on
clear assumptions and rules?" Status might be lost, but I feel that
clarity would be gained.
I feel that we have to check, and doubt, even when that's neither politic,
nor pleasant, nor easy. That would argue, and argue strongly, for the
experimental mathematics position of Godel, Chaitin, and others. It would
also cast a stark light on the reasons why Godel's work had so little
practical effect on Hilbert-inspired doctrine and usages in mathematics.
(See the first sections of "Randomness in Arithmetic and the Decline &
Fall of Reductionism in Pure Mathematics" by G.J. Chaitin
http://www.cs.auckland.ac.nz/CDMTCS/chaitin/unm.html)
*******
I appreciated Michael Detlefsen's comparison and contrast of Hilbert's
formalism and Russell's logicism, and found it interesting and clarifying
in many ways. But at the level of fundamentals, I sympathize with
Gordon Fisher's strong, sharp arguments for human limits. Fisher stands
against claims of certainty and rigor that have been socially and
academically important over many decades. These are claims that have
shaped many careers. Those claims must logically depend on attributions
to human being of powers that all human beings lack.
M. Robert Showalter