Talk:Where Mathematics Comes From
the quotes were already in fair use on the page of reviews of Lakoff and Nunez referenced at the bottom - I presumed (perhaps wrongly?) that this means that they were cleared for quotation. Only the Santa Fe one seemed to be long, but it's quite specific, and justifies the claims int he rest of the article, so I'm not sure the scope of this is very clear without it... but I'll go with consensus, obviously.
It's significant to the article that "counting up to four" and moving along a line are empirically observed cognitive phenomena. Does it really make sense without that?
The Sigma Xi review is of the first edition which had technical errors.
My careful read of the other reviews, which include some pretty prominent journals and institutes, didn't make note of those errors, and the reviews have been up for some time, so presumably they'd object if they thought the overall theory was wrong.
Nonetheless, it speaks to the care of the authors that the technical errors be mentioned by some neutral party, so the Sigma Xi review belongs there - maybe with a note to the effect that these errors don't seem to have caused the other later reviewers to give up on the theory.
OK- scratch that - I see there's now a link to their "warning" about this.
Removed this from the article:
- It may well be that turning mathematics into an empirical science will involve a great deal of animal testing, to determine what's shared - and what is merely a widely shared human bias, arising out of our over-complex brains.
Many things may be, but this is unlikely to be one of them. Who says this? Other than you?
The objection is legit, thanks, but when you changed it you said this was deleted for being "surreal" - I admit it's speculative, but what's "surreal" about empirical testing of a cognitive science thesis, to see what we say share with apes and what is uniquely or bizarrely human?
Most of anthropology and primatology recently seems to be testing what things apes can do, what they can't, where we share a foundation ontology with them, where we don't. For instance how do they see 'family', or 'friend', or etc..
I also can't be the first person to call the human brain "over-complex"... I thought it was kind of a crack on the whole community arguing this stuff, as well... many wouuld just say "mathematics works" and leave it at that...
But the article is controversial enough without this suggestion of a path to validating... I'll actually see if I can get a quote out of Lakoff or find the material on chimps being tested to determine who real "number" is to them - saw this being done in a lab in Japan - on the Discovery channel - as usual the credits scroll by too fast adn the researchers name is too Japanese. ;-) But I'll dig it up.
"Clearly, when a man shoots a bear it is not only the man whose experience of the bullet is defined by "F=MA"."
This was actually the exact sentence (in private converation) that convinced me that mathematics could not be wholly a human invention...
The bit about nuclear weapons seems to be totally unrelated to the topic in the first and second sentences of that paragraph. The article as a whole is a little confused and poorly organised - perhaps a re-write is in order? As to animal testing... You mean experiments conducted using animals which is an different kettle of fish. -- The Ostrich
No, both topics are related, and if that's not clear, I'll fix it. Tell me if it makes more sense to you this way. Lengthy but clear:
1. if mathematics is a system that arises from constraints in the cognitive makeup of humans, then we cannot know what is "human delusion" and what is "objectively real" without some non-human animals to test mathematics on. Scientists in Japan are presently testing chimps to see how much of math they can master. If it turns out that they can master all the basic traits like "counting up to four", then this cognitive science of mathematics must apply to them as much as to humans - and we ought to be able to discuss it with them, or jointly agree with them on concepts like "whether this is four coconuts or not". This amounts to a primate testing of mathematics itself.
It is certainly an experiment, and it is certainly conducted using animals, and it is being done now. It's a glaring and obvious issue with the L&N claim that somehow mathematics is "uniquely human" or that we "can't know how much of it is objectively real" - we can at least know how much is shared with near cousins.
2. more difficult, the physics question. Addressed somewhat in particle physics foundation ontology. Tom Siegfried's objection is different, that what scientists see in a particle accelerator can be modelled using math - although Dirac had to invent a different notation I think that's a side issue.
If mathematical models such as those in physics are shared only by humans, and it's not clear that the "reasons why we believe there is a new particle" can be shared beyond humans, i.e. the chimps don't know what we're talking about, and when they look at the charts they just scratch their heads like any untrained human, then the reality of these theories are on shaky ground.
If it's only highly trained humans saying that they saw this particle and that this math is therefore "real", well, we start to be on shaky ground... Siegfried's assertion may well be more controversial than any by L&N here.
3. most difficult, the ethics question. The use of nuclear weapons and particle accelerators are restricted by a lack of opportunity to test the theories. WE CANNOT SIMPLY TEST ALL THEORIES OF WHAT WILL HAPPEN EQUALLY - therefore we have a lack of objectivity in experiments the same as we (might, if you accept this cogsci of math stuff) may lack such objectivity in notation.
This is very closely related to the argument about censoring science distorting it, and the Precautionary Principle which says you should not test a theory if one of the conceived outcomes of the test is destroying something you can't replace. Probably this is too complex to bring up in this entry, but:
We have ethical obligations not to destroy the planet to see if nuclear winter will happen, and there is a limited amount of particle acclerator time for which scientists compete fiercely. So inter-human politics absolutely deterines what theories get tested, which tends to determine what theories get discussed, which is in turn going to guide what theories get proposed.
This is also called "the paradigm problem" - when do we give up on some infrastructure, and tell the scientists who were "improving" it or using it "to test a theory" to go home, and that they aren't needed any more....
Extreme form of this argument: nuclear weapons aren't needed since they lead to more trouble than they're worth. Therefore, particle acclerators aren't needed since they are likely to shed light only on more ways to get big bursts of energy and blow up more stuff at once, or make black holes to suck up the Earth. Therefore, particle physicists aren't needed, etc...
If you accept the cogsci of mathematics as real, then all this stuff that was built assuming that the mathematical models were "real" and could be validated by simulations (purely mathematical) or complex webs of assumptions about the observer infrastructure, becomes very very doubtful.
So, it's one of the objections to this, that the cogsci of math is just one of those politically motivated theories that gets in the way of "real science"
I wanted to deal with that objection, which isn't possible without laying it out a bit.
Whew.
OK, now I need a beer.
I think the "bear line" and the "primate testing" line really get across the point. However, it would be nice not to have to lay out ALL the details of the objection and debate as above.
I also still think that without the Santa Fe quote, no one is going to really understand the scope of this, or how limited it really is - making it feasible to test - even on chimps.
OK, I see what you mean about the nuclear weapons thing, it wasn't clear that this is a view of Zerzan and Waring rather than Wigner. Now it is, and there's some bridge there.
I also mentioned the relationship with the Precautionary Principle - and the common theme of limiting trust in human constructed mathematical models, and deliberately choosing "not to go there" as a consequence of nuclear standoffs.
I could have mentioned climate change too, but that's not generally seen to be something that can destroy all cognitive beings - just make us war more...
Lakoff is a highly political commentator so it's not really right to avoid the politics of his conclusions. Although he carefully quotes in his reviews people from technical communities who agree with him, his overstatement of the degree to which "mathematics is human" is a clear sign of a certain bias.
Mathematics, to the degree he's talking about it, which is not very far, is just as well understood by chimps, and probably by dolphins, horses, and dogs.
OK, I added headings that at least ask what I think are important questions.
I didn't answer the questions, really, just laid out some controversies.
Ultimately this article has more to do with the difference between ethics and morals, and the Neutral Point of View, than most other topics in the sciences.
Maybe it should link to some stuff in the meta?
OK, a bit of research has determined that this topic isn't entirely crackpot (assuming that Berkeley doesn't make crackpots professors) so for those people like myself who would be tempted to delete this article straight away as content-free ravings, don't.
However, the article seems to me to be infested with postmodernists syndrome - the inability to simply and clearly explain WTF they are on about.
Could the article please *clearly* state:
- What this theory states.
- Why the proponents believe it.
- What connections it makes with other ideas (origins etc.) Frankly, it sounds like the kind of tripe Sokal parodied.
- Who agrees with them.
- Who disagrees with them.
- Is it an active area of further work? By whom?
- Whether any actual mathematicians give a rat's.
"What this theory states" was deleted because someone was worried the quote, which was already used on L&N's own review page, was too long and copyrighted. I have stated several times it should come back. So I'll bring it back.
"Why" is anybody's guess, but I suppose it's because they came up with a rationalization that did not seem to get any major mathematicians angry... and a lot of them appear to have had a crack at the material, looking at that review page
"Connections"? Well, it's basically asking if math is really a NPOV, or if it reflects domination desires by humans over nature, extensions of things that work to blow stuff up and build infrastructure into control of humans... a lot of the issues that indigenous peoples tend to take up with colonials...
"dis/agree" - hard to say - it seems some reviewers question the implications or to what degree this is true of humans vs. apes or vs. robots or vs. dogs, but no one doubts that *some* of mathematics is basically just like numerology - sophistry that adds up internally but applies to nothing else.
I've been looking for a strong attack and it just isn't there - Lakoff is *very* respected and most people consider him the heir to Chomsky and the prime theorist of language.
L&N, and Santa fe inst., claimed it was "an extended start on a cogntiive science of mathematics" and I know the work is continuing in private and on mailing lists. There is some work on cognitive political science for instance.
Mathematicians caring wouldn't really be the point - a mathematician only cares if his work is internally consistent, not what it externally predicts or describes. Other scientists, especially physicists working on expensive infrastructure available to few people, would care more, maybe oppose more...
Mathematical epistemologists, and "foundationists" would be the people to ask.
OK, hopefully this is more structured, and puts some context in there. This is a paradigm shift, a huge program, like Principia Mathematica (1913, Russell and Whitehead)... like that program it could fail. But it's as big as that was, actually bigger.
I hope I captured the implications... without seeming to advocate any of 'em.
This is not perfect. I should ask Lakoff himself to look it over - and grant permission maybe for more extensive quotes. I don't think seven or eight lines that is basically a summary of the work as it would appear in a scientific abstract is a copyright problem - but who knows what lawyers believe?
he might also outline where the work is going next, who's doing it, and etc.
OK, I just sent the article to Lakoff, with the URL. He may edit it here, or he may respond to a list... either way I ask that people leave this alone for a week or so. Thanks.
--- This was just removed:
"[science is an ambiguous project]...One which we may well abandon.
It seems to be an ethical choice: accept a moral instinct, or engage in some ethical negotiation with some other body to gain resources to test new models.
Simply proving a model does not contradict other models may no longer be good enough to convince funders that every step of the derivation was "embodied" - that it relies on axioms and proofs that are themselves completely grounded. "
I wonder if the editor is basically practicing an ideology of "scientism", whether he knows it or not: assuming scientific knowledge not to be subject to ethical choices, e.g. to fund or not to fund, to experiment or not to...
The article makes sense without these questions being raised, but the ethical point about the practice and funding of science is now completely excised... and it's a pretty major point.
- Is this a major point that Lakos makes, or a major point that is dear to your heart and that you somehow want to weave in? AxelBoldt
If it's a philosophical issue, maybe it's another article. There are lots of writers now talking about "the death of science" and such... including the head of the AAAS who used that phrase to describe censorship of anthrax stuff.
If it's a matter of bad writing, ok, guilty, I can go back and put these concerns more cleanly elsewhere. The idea of viewing the body as an axiom base was also removed... not surprising, as this seems to also have been the issue between Lakoff and his reviewer from the MAA - one of the links at the end. Is " bodies as theses? " a fair summary title for a paragraph on this?
I believe that among technically literate people, there is a general consensus that mathematics is a neutral point of view, indeed that if logic itself is a valid mode of investigation, mathematics must equally be one. I therefore think that this article should at least refect these facts. Perhaps we should get Alan Sokal to give this one a re-write? -- The Ostrich.
That's exactly the point... that "technically literate people" assume exactly that. This analysis challenges that. If it doesn't clear define and reflect what it is challenging, that's a flaw in the introduction.
How about simply adding exactly that statement of yours near the beginning?
The issue of "logic itself as a valid mode of investigation" and assumptions about all of mathematics necessarily sharing that, is what's called a "foundation issue" in mathematics.
Perhaps a summary of Whitehead, Frege, Russell, and Godel near the beginning of this, and the present lack of agreement on what mathematics means, is required as well.
I'd suggest also that "technically literate people" are not the audience we are writing for, necessarily, and that this article addresses conflicts between a 'neutral' and 'natural' point of view (being discussed in the meta).
BTW, I consider myself "technically literate" in exactly the sense you mean, with degrees in Mathematics itself, and I do not consider something stated in mathematical or logical or set theory form to say anything about the geometry it measures.
There is a very well founded literature in cogsci of 'what measurement is' - Tversky, Kahneman, and others laid the foundation for this work in the 60s, so its not coming from left field.
Admittedly, it's a hard topic to introduce to someone unfamiliar with this material, or with a strict Western/dualist view of subject/object relations.
OK, done, to the best of my off-the-cuff ability. Does that read reasonably? It sure outlines a lot of other stuff that needs to get included...
The choice of reactions to the first nuclear explosion helps highlight the fact that humans define their own tests for what "works", and therefore that their shared belief in mathematics isn't necessarily a belief in more than their own cognition and culture.
That seems to be Lakoff's claim, based on his "there's no way we scientifically could possible tell".
Thanks for your contributions. This article is now taking shape nicely, and moving towards NPOV.
I don't agree with many of the statements in the article, but as they become increasingly attributed to their sources, as facts that 'X says Y about Z', this does not matter. Other authors can add their notes of the contrary views of others in the same format, and the Wikipedia process will polish the article.
From the article:
- Alfred North Whitehead, Bertrand Russell, and Kurt Godel established that logic and set theory were in some sense grounded on something else, something geometric and quite "real",
Please explain: Russell showed that naive set theory led to a paradox, Godel demonstrated that formal axiomatic systems with enough power to do arithmetic cannot be both consistent and complete. I'm not sure that either result implies the second half of the sentence. The Anome