Comments on WHERE MATHEMATICS COMES FROM

L.A. Loren lloren at mitre.org
Fri Jan 19 10:13:43 EST 2001


At the outset I should point out that I haven't had a chance to read the
book yet. Although after talking to a couple folks who have read it I am
inclined to think that you are misrepresenting the author's arguments
here. Also, you lapse into several fallacies during the course of your
critique. Since I have a few minutes to spare lets take a quick look.

Steven Michael Harris wrote:
> 
> Comments on WHERE MATHEMATICS COMES FROM
> (HOW THE EMBODIED MIND BRINGS MATHEMATICS INTO BEING) by George Lakoff and
> Rafael E. Núñez
[snip]
> 
> The arrogance of the authors' statements is significant. They put forth two
> questions:

I'm not sure where the "arrogance" is, but you are dangerously close to
an ad hominum here.
 
> 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!

Now you're just insulting the authors and I have no idea what it is you
object to. It looks to me like the authors are correct so far. "what
cognitive mechanisms are used to formulate mathematical ideas" does
appear to be an empirical question, and an interdisciplinary one at that
(neuroscience, psychology, etc.)
 
> 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.

First, I have no idea who "they" are. Second, nobody can agree on what
consciousness is. Third, who is this "they" who claim to have the only
approach for finding answers (apart from the obvious fact that everyone
believes they are on the right track while others in their field are
not). Finally, in the preceding section you attacked cognitive
scientists, now you are after philosophers. Perhaps you would do better
to criticize the authors rather than every larger group to which the
authors might belong.

[snip]

> 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?

The question is "what *cognitive mechanisms* permit people and animals
to make mathematical calculations". I fail to see what insights
mathematics might have to offer. Math is really good to know, but I
don't think it will tell you very much about cognitive mechanisms.

> 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.

Again, you are bordering on an ad hominum here. You may not agree with
the authors, but they are reasonably bright people. Although they might
be mistaken about it, I'm pretty sure they haven't overlooked the
connection between language and mathematics (again not having read the
book I can't say this for certain, but if I had to bet on it I'd say
they are aware of this connection).
 
[snip]

> I'll need to quote extensively
> from this chapter in order to present a fair argument.

Finally you are attempting to present a fair argument.

[snip]

> 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. 

This is the first well reasoned point you have made.

> 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.

you might want to avoid this line of argumentation. Nobody is going to
be swayed by the claim "I have a better way of explaining this...but
it's too complicated to go into".

> 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. 

Once again, I have not read this book but in light of some of the
authors previous works I think it is unlikely that they maintain this
position.

[snip]

> 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).

There are mathematical equations that describe planetary motion and
allow us to make predictions, but these equations do not explain
planetary motion. Similarly, there are equations that describe the
interception of a moving object, but it does not follow from this that
these same equations are being used to accomplish the task of
intercepting a moving object (they may in fact be used, but it requires
additional argumentation and does not follow directly).

> 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.)

This is the kind of argument you should have been providing all along.
 
> You only need three points of observation on the arc of a ball to predict
> where it is going to land.

Actually it can be done with as few as two under the appropriate
circumstances, but this is just a nit.

[snip]


OK well, I'm done with my morning coffee so I really should start to
work.

Cheers,
Lew






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