Comments on WHERE MATHEMATICS COMES FROM

Steven Michael Harris stevenharris at mediaone.net
Fri Jan 19 07:11:52 EST 2001


Comments on WHERE MATHEMATICS COMES FROM
(HOW THE EMBODIED MIND BRINGS MATHEMATICS INTO BEING) by George Lakoff and
Rafael E. Núñez


...WHERE MATHEMATICS COMES FROM provides a theory built entirely on an
assumption of proof that there is an innate and simple mathematics built
into our brains and into the brains of some other animals based on some
flimsy "proof."

The arrogance of the authors' statements is significant. They put forth two
questions:

Exactly what mechanisms of the human brain and mind allow human beings to
formulate mathematical ideas and reason mathematically?
Is brain-and-mind-based mathematics all that mathematics is? Or is there, as
Platonists have suggested, a disembodied mathematics transcending all bodies
and minds and structuring the universe - this universe and every possible
universe?
Then they answer their own question:

"Question 1 is a scientific question, a question to be answered by cognitive
science, the interdisciplinary science of the mind. As an empirical question
about the human mind and brain, it cannot be studied purely within
mathematics. And as a question for empirical science, it cannot be answered
by an a priori philosophy or by mathematics itself. It requires an
understanding of human cognitive processes and the human brain. Cognitive
science matters to mathematics because only cognitive science can answer
this question."
This statement demands a strong rebuttal. What a piece of crap!

In the current state of cognitive science the philosophers of consciousness
can't agree on a definition for the word "consciousness." They don't know
what cognition is and then they claim that they have the only approach for
finding the answers about cognition.

Even though the authors are focusing on language-based-brain-and-mind-based
mathematics when they talk about brain-and-mind-based mathematics, they do
much better when they answer the second question:

Theorems that human beings prove are within a human mathematical conceptual
system.
All the mathematical knowledge that we have or can have is knowledge within
human mathematics.
There is no way to know whether theorems proved by human mathematicians have
any objective truth, external to human beings or any other beings.
Unfortunately they then start to corrupt this logic by referring to the
incomprehensibility of Platonic math being akin to the incomprehensibility
of God. Never trust "science" that brings religion in to support the
argument. Such an approach never brings clarity.

In arguing their point they say that "it is only through cognitive science -
the interdisciplinary study of mind, brain, and their relation - that we can
answer the question: What is the nature of the only mathematics that human
beings know or can know?" Wouldn't an "interdisciplinary" study of
mathematical thinking include the discipline of mathematics itself as a
contributor with possible insight? Especially when none of the fields of
study has yet come up with an answer? Especially when the current
mathematical approach to biology and behavior is so vague and inaccurate in
spotting patterns?

They would be right when the say "that human mind-based mathematics uses
conceptual metaphors as part of the mathematics itself" if they had the
insight to know they were really describing language-based human mind-based
mathematics.

Their next conclusion that "human mathematics cannot be a part of a
transcendent Platonic mathematics, if such exists" is just not provable.

They don't consider the possibility that their observation that a
"conceptual metaphor is a cognitive mechanism for allowing us to reason
about one kind of thing as if it were another" exists because the
mathematical properties of the non-language based thinking are consistent
across the different realms of processing and perception.

The Brain's Innate Arithmetic?
The entire argument they present is based on the "proof" that there is a
basic "hard-wired," genetically programmed ability to perform some simple
math (language-based math) in some animals and in humans before any such
ability is taught. Most of this proof is presented in Chapter 1 and the rest
of the book is pretty much a synopsis of all mathematics, or the evolution
of mathematics, constantly referring back to the proof presented in the
first chapter. So the first chapter is where most of my disagreement will be
based as well. It will not take that long.

They claim number discrimination by pre-language babies using some suspect
logic.

They state as fact that "babies have the following numerical abilities":

At three or four days, a baby can discriminate between collections of two
and three items (Antell & Keating, 1983). Under certain conditions, infants
can even distinguish three items from four (Strauss & Curtis, 1981; van
Loosbroek & Smitsman, 1990).
By four and a half months, a baby "can tell" that one plus one is two and
that two minus one is one (Wynn, 1992a).
A little later, infants "can tell" that two plus one is three and that three
minus one is two (Wynn, 1995).
These abilities are not restricted to visual arrays. Babies can also
discriminate numbers of sounds of two or three syllables (Bijeljac-Babic,
Bertoncini, & Mehler, 1991).
And at about seven months, babies can recognize the numerical equivalence
between arrays of objects and drumbeats of the same number (Starkey, Spelke,
& Gelman, 1990).
The assumptions are that the abilities are of a mathematics (language-based
mathematics) similar to what we use when we count in our heads "one, two,
three." but these assumptions come from understandable language-bias in
interpreting observations from various studies. There are other ways to
interpret the observations in these studies. I'll need to quote extensively
from this chapter in order to present a fair argument.

"Slides were projected on a screen in front of babies sitting on their
mother's lap. The time a baby spent looking at each slide before turning
away was carefully monitored. When the bay started looking elsewhere, a new
slide appeared on the screen. At first, the slides contained two large black
dots. During the trials, the baby was shown the same numbers of dots, though
separated horizontally by different distances. After a while, the baby would
start looking at the slides for shorter and shorter periods of time. This is
technically called habituation; nontechnically, the baby got bored."
"The slides were then changed without warning to three black dots.
Immediately the baby started to stare longer, exhibiting what psychologists
call a longer fixation time. The consistent difference of fixation times
informs psychologists that the baby could tell the difference between two
and three dots. The experiment was repeated with the three dots first, then
the two dots. The results were the same. These experiments were first tried
with babies between four and five months of age, but later it was shown that
newborn babies at three or four days showed the same results (Antell &
Keating, 1983). These findings have been replicated not just with dots but
with slides showing objects of different shapes, sizes, and alignments
(Strauss & Curtis, 1981). Such experiments suggest that the ability to
distinguish small numbers is present in newborns, and thus that there is at
least some innate numerical capacity."
They later state that the innate ability to count to three and sometimes to
four is present at a very early age. But none of these studies has proved
that counting (language-based math) has occurred. There are a variety of
ways to approach this. It is more complicated to explain because innate
biological math is immensely complicated in all animals with enough of a
nervous system to be animated in complicated ways or to have any evolved
sense of sight or hearing or to have a liver or.

One of the problems in this kind of logic is the assumption that the brain
of any organism is completely the product of genetics, of a hard-wired
pre-ordained ability like that of a machine. But a nervous system is an
organization of logic that forms itself based on the mathematical principles
it uses to operate and is affected by genetics but really operates as a
mathematical logic that is constrained by the limits imposed by the
genetics. Otherwise it would not be possible for a damaged brain to flexibly
rearrange its logic in order to repair a function damaged by lesion or
excessive cerebral fluid or whatever by using another region of the brain or
another arrangement of the cells (following the logic of how brain cells
interact with other cells).

Forget for a moment the fact that any animal that can intercept a moving
object or that can calculate information based on arrangements of light
affecting cells in one part of the body and calculate the necessary
movements of thousands of muscle fibers in order to predict the existence of
an object away from the body and reach out and find that object is using
massive mathematical calculations to perform such feats (even though there
is no language-based understanding of such math).

When the baby responds to a change from one to two, from two to three,
rarely from three to four, but never from four to five - something besides
"counting" is going on.

Think of it this way. The baby is reacting more significantly to observable
change. The difference between one and two is a 100% change. The difference
between two and three is a 50% change - still a significant change in amount
or degree. Beyond three the amount of change is a minority of change. From
three to four is a change of 33% and from four to five is a 25% change in
amount, so the response to such change is less likely as there is a much
smaller percentage of change. (If somebody gave you a glass with milk in it
and added 25% to it when you were not looking, you might not notice the
difference.) So this does not necessarily represent counting ability. (They
never said if the study also tried to see a difference in response from
three to five - a 66% change in amount.)

(Another way of saying the same argument: if you are napping and I increase
the light in the room 100%, you are much more likely to respond by waking
than if I just increased the light in the room by 25% or gradually increased
the light by 100%.)

Another way of looking at it is that beyond three the brain might be using
shorthand to assume further repetition so it does not have to continually
process repeating objects. Remember that it might just take three
observations of repetition to recognize a predictable pattern and after that
an assumption of repetition might be the impulse. (The brain predicts
objects - especially repeating objects such as the pattern in tile or
wallpaper when filling in the blind-spot in vision, the reason that you can
put a unique object into your blind spot and it disappears when repeating
visual patterns surround the blind spot.)

You only need three points of observation on the arc of a ball to predict
where it is going to land.

Any quantity beyond three might be inherently boring or just more than is
needed when a pattern needs to be perceived.

In comedy, everything is setup in threes. The costume will always have three
buttons, never four (unless when created by an amateur). The jokes are setup
in threes. There are always three people walking into a bar. The anecdotes
concerning the first two people setup the pattern. The third anecdote would
confirm the pattern but in comedy there is always a switch in the pattern
with the third anecdote which creates the humor. A joke that has four people
walking into a bar would lose the audience if the punchline only came with
the fourth anecdote (but would work if the switch occurred with the third
person and then another switch was delivered concerning the fourth person).

It is our nature to assume that a pattern will go on indefinitely when
established with three consistent examples of the pattern. (The ellipsis is
three dots.) (Infinity is represented by three dots in mathematics.) This
point is even mentioned by the authors in this book in a later chapter when
talking about a different subject (making this essay of mine so much
easier):

"Consider a sentence like John jumped and jumped again, and jumped again.
Here we have an iteration of three jumps. But John jumped and jumped and
jumped is usually interpreted not as three jumps but as an open-ended,
indefinite number.".
"But verbs like swim, fly, and roll are imperfective, with no indicated
endpoint. Consider sentences indicating iteration via the syntactic device
of conjunction: John swam and swam and swam. The eagle flew and flew and
flew. This sentence structure, which would normally indicate indefinite
iteration with perfective verbs, here indicates a continuous process of
swimming or flying. The same is true in the case of aspectual particles like
on and over. For example, John said the sentence over indicates a single
iteration of the sentence. But John said the sentence over and over and over
indicates ongoing repetition. Similarly, The barrel rolled over and over
indicates indefinitely continuous rolling, and The eagle flew on and on
indicates indefinitely continuous flying. In these sentences, the language
of iteration for perfectives (e.g., verb and verb and verb; over and over
and over) is used with imperfectives to express something quite different -
namely, an indefinitely continuous process.".
As a writer I would have chosen to assume I'd made my point earlier and
moved on (because the joke should be over with the third example), but these
authors (in the same way they repeated their points in other parts of the
book) kept going on and on and on.




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