User blog comment:Alemagno12/BM2 has a problem/@comment-30754445-20180724073311/@comment-30754445-20180726085642

@P進大好きbot

"The word 'evidence' is sometimes used with the meaning 'it looks/should be true' "

Precisely.

But the googolplex dollar question is: How do we know when a mathematical statement "looks true"? What's the difference between "wild wishful thinking" and a conjecture which should be take seriously?

You are, of-course, 100% right that a conjecture remains a conjecture until it is regoriously proven. But this does not change the fact that not all conjectures are equally suspect. Goldbach's conjecture, for example, is very very very likely to be true. The fact that we don't have an actual proof does leave a shred of doubt, but still... it would be counterproductive to equate such a conjecture with baseless wild speculation.

So how do we tell the difference? What constitutes "evidence" (with the word-usage you've described) in mathematics?

This is a very complicated issue. But one thing is clear: A simple table - no matter how long - does not constitute evidence of any sort. The reason for this is that:

(1) There are an infinity of numbers, so any finite table will cover precisely 0% of the cases.

(2) Some patterns may take a very very very long time to emerge.

Now, a table can be immensly useful, when combined with some theoretical reasoning (attention to those who claim I'm demanding a regorous proof for everything). I'm simply saying that longer tables don't necessarily translate into a more persuasive argument.

BTW any topic of googological interest, by its very nature, would also be an example of this. Does {3,3,3,3} in BEAF ever terminate? Does the Goodstein Sequence starting with 4 ever reaches 0? Can we write a TREE(3)-esque sequence of trees that goes on forever?

Trying to brute-force our way through these questions, we will get the wrong answer for every single one of them. Even if we fill the entire universe with our step-by-step process, we'll never finish.

This, right here, tells us another important thing: When creating large tables of data for evidence, the precise nature of the data matters. Googologists, of all people, should be aware of this difference between quantity and quality when it comes to doing analyses.

"Before recalling such a false conjecture, you can understand what I meant by seeing my last example, i.e. the Euler's function."

I don't like the example of Euler's polynomial for "evidence can be misleading", because:

(1) Getting your first counter-example after a mere 40 tries is hardly mind-boggling.

(2) The number 41 is right there in the equation itself. After all, you wouldn't be surprised to learn that the diophantine equation 3X=2764584994246624329 has a very big number as its first (and only) solution, would you?

The situation with Gauss' Li(x)/Pi(x) conjecture is far more surprising, which is why I've chosen to give it.

Or, if you prefer an example with simple polynomial equations, how about Pell's equations? For example, consider this diophantine equation:

x2–109y2 = 1

Does it have a solution?

Given that 109 appears in the equation, most people wouldn't be too surprised if the first solution was in the hundreds or the thousands. But after testing a billion cases, they'll become pretty convinced that there's no solution... right? (the smallest solution has both x and y in the trillions)

(as a side note: I love Euler's polynomial for other reasons. The fact that it works for n=0 to 39 is a result of a really nifty bit of number theory. It's beautiful. It just isn't a very fitting example for the topic we're discussing here)