User nick thompson - MathOverflow

Answer by Nick Thompson for Demonstrating that rigour is important

2011-05-04T13:42:07Z

This is not strictly an answer to the question since it is a hypothetical rather than an example, but perhaps relevant nonetheless: Suppose that you have some computer program-say for keeping an airliner stable under gusts of wind-and it relies on a numerical algorithm, proved to converge under reasonable assumptions about air pressure and wind velocity, so on. A faster and more stable algorithm is developed for which no proof of convergence is known, though all the researchers in the field assure you that it always converges and that they are certain that it will always converge given the plausible scenarios your code is likely to be used for. I think it is clear that you should not trust their judgment, but rather retain the old code, despite the clear desirability of having a faster numerical algorithm.

So from the perspective of the researchers, a proof might not be all that important; it may have only told them what they already knew and generated no new insights in the process. But from the perspective of the consumers of mathematics, knowledge that a proof exists may lead to incremental improvements in technology that would otherwise not happen. Of course, at this point we've come full circle and it becomes important to the researchers to supply a proof.

My second point again distinguishes between consumers of mathematics and researchers: It is sometimes much easier to become 100% certain of something than 99% certain. 99% certainty that a given statement is true requires thinking about many concrete examples and developing intuition, whereas 100% certainty requires logically assenting to the statements contained in a proof. By this standard, I would say that I am 100% certain about the bulk of my mathematical knowledge, and not 99% certain. Perhaps this is a lamentable state of affairs, but time is finite. We cannot hope to develop intuition about all the statements we wish to use while working on problems we do wish to develop intuition about. In that sense, proofs encode in a few kb's the vast amount of information stored as the intuitions of all the researchers working on a particular problem. Again under this model, the purpose of proofs is for the convenience of non-researchers.

Answer by Nick Thompson for Defining variable, symbol, indeterminate and parameter

2011-04-27T09:54:17Z

In written English (and of course other languages), we have linguistic constructs which tell the reader how to approach the ideas that are about to be presented. For example, if I begin a sentence with "However, . . .", the reader expects a caution about a previously stated proposition, but if I begin the sentence with "Indeed, . . . ", the reader expects supporting evidence for a position. Of course we could completely discard such language and the same ideas would be communicated, but at much greater effort. I regard the words "variable", "constant", "parameter", and so on, in much the same way I regard "however", "indeed", and "of course"; these words are informing me about potential ways to envision the objects I am learning about. For example, when I read that "$x$ is a variable", I regard $x$ as able to engage in movement; it can float about the set it is defined upon. But if $c$ is an element of the same set, I regard it as nailed down; "for each" is the appropriate quantifier for the letter $c$. And when (say) $\xi$ is a parameter, then I envision an uncountable set of objects generated by $\xi$, but $\xi$ itself cannot engage in movement. Finally, when an object is referred to as a symbol, then I regard its ontological status as in doubt until further proof is given. Such as: "Let the symbol '$Lv$' denote the limit of the sequence $\lbrace L_{n}v \rbrace_{n=1}^{\infty}$ for each $v \in V$. With this definition, we can regard $L$ as a function defined on $V$. . . "

So in short, I regard constructing precise mathematical definitions for these terms as equivalent to getting everyone to have the same mental visions of abstract objects.

Comment by Nick Thompson

2011-04-28T03:02:37Z

I believe that "variable", "constant", and "parameter" have identical set theoretic meaning, as they operate as adjectives describing elements of sets within any given proof, and the validity of proofs depends only on the properties of the elements of the sets under consideration, not the adjectives used to describe the elements. So though we regard "variable" as a noun, it arises from the mental abstraction of an adjective. Objects which seem to be amenable to precise mathematical definitions seem to arise as abstractions of nouns. (That's the best answer I can come up with unfortunately!)