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http://www.boston.com/bostonglobe/ideas/articles/2009/02/08/a_talk_with_mario_livio/?page=1

In the interview Livio claims that math is simply a human invention because:

"Let me start with this silly idea - the isolated jellyfish. Imagine that all the intelligence resided not in humans, but in some isolated jellyfish at the bottom of the Pacific Ocean. This jellyfish - all it would feel would be the pressure of the water, the temperature of the water, the motion of the water. Would this jellyfish have invented the natural numbers - 1, 2, 3, 4, 5, and so on?"

His general claim seems to be that math is a human invention because: "Humans at some level chose the mathematical tools based on them being suitable for the particular problem." And when asked "How is math different from other human inventions, like art?" Livio answers: "Mathematics is somewhat special in that it has an incredible longevity."

There is an obvious misunderstanding in his argument - there is a difference between making up something and discovering only what you care about. Whether Jellyfish would come up with a different chunk of mathematics is the same, by his logic, as whether a Russian and an American would. Or simply different people have different interest and are drawn to different questions and so will solve different questions. Obviously. An algebraist would have a different math, so to say, than an analyst. This question has no bearing whatsoever on whether math is true and whether it is objective. We find out the chunks we need but his argument accepts it all as a discovery.

Moreover his argument seems especially stupid coming from a scientist. To ask whether the scientific rules are true or not is a very different question from whether if we were living underwater we would uncover other theories before uncovering Gravity. The order and interest of the theories have nothing to bear on the basic question.

In any case his arguments in some form could have been interesting 150 years ago, but he's a bit late. When it was discovered, more than a 100 years ago, that there are other kinds of geometries and not simply Euclidean that was revolutionary. Math just appropriated the different kinds into it, and now all of them coexist inside math. It is not whether math is true or not anymore. Even finding different axiomatic bases for mathematics where it only works inside them, even that was appropriated into math.

Could one ask whether math is true? Yes. One can ask whether 2+2=4 or whether it is a manner of agreement, and perhaps 2+2=5. The more one really sees how things operate in the mathematical world the more one feels comfortable asking this. Of course the arguments for and against would have nothing to do with Livio's arguments.

Sidenote: the article starts by adding a quote from a "Nobel Laureate," in a manner which reminded me of Layla's recent post on quoting.

(**) (sorry, guess that was a crucial step in the proof..."then a miracle occurs.")

I find it hard to understand how mathematics can be anything *but* a human invention. A mathematical argument can only be consistent or inconsistent with some set of axioms, which are, by definition, assumed (by humans, who else?) That is the meaning of "true" and "false" in mathematics. To say that 2+2=4 is true is to say that it is consistent with the definition of 2, 4 and +. The only relation of a mathematical entity to something that can be considered *really* "true" (without getting into the question of what is really true) is when it is applied to reality by some scientific theory, in which case we would ask whether that scientific theory is true or false. Which is why I feel uncomfortable with referring to mathematical progress as "discovery". One discovers things that were already there but he was unaware of. A mathematical theory can be considered a discovery only in the sense that its consistency (with other mathematical branches, or within itself) has been proven, and thus "discovered". The theory itself is invented. It does not exist in the same way that gravity, the atom or America exist. Scientific discovery is limited to what exists, while mathematical "discovery" is limited to all that is not contradictory. To argue that these two limits are equivalent is far from trivial. Intuitively, it seems to me that the latter is infinitely greater than the former.

As for the interview -- I don't know if this is the case here, but bear in mind that journalists often distort what they report in ridiculous ways.

As for the interview -- I don't know if this is the case here, but bear in mind that journalists often distort what they report in ridiculous ways.

Damien, to clarify, when you say "one discovers things that were already there but he was unaware of," do you mean that this is what discovery properly is, or that this is what mathematicians purport to do?

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