Take something like sigma summation (∑), at first glance it look like an abstraction, but it’s too leaky to be a good one. Why is it leaky ? Because all the concept it “abstracts” over need to be understood in order to use the sigma summation. You need to “understand” operations, operators, numbers, ranges, limits and series.
It’s just a for loop.
As long as we are willing to trust the implementation of the software and hardware, we can quickly validate any mathematical tool over a very large finite domain.
There’s a smaller domain where we can validate the proposed counterexamples. To borrow from your example: “this is obviously false” 27^5+84^5+110^5+133^5=144^5.
Obviously there’s cases where this concept break down a bit (e.g. Ramanujan summation), but these are the exception rather than the rule.
How is this a break down? (I don’t know what you’re building.)
They have a lot of tweakable hyperparameters and they are no modifiable just in principle.
It’s not a for loop, for loops don’t deal with infinity as far as I know.
How is this a break down? (I don’t know what you’re building.)
As in, the results 1 + 2 + 3 + 4 … = −1/12 is “obviously false” yet mathematically true. So the pattern of “if something is true in a very intuitive way than it must be mathematically true”, doesn’t hold in those kind of cases (as opposed to the 2 * 3 = 6 case, where mathematics correctly describe what we intuit to be true by saying the statement is correct, at least if you think of mathematics as a “language” in the “programming but running on wetware” sense.
I’m pretty sure appending a single number to an infinite series is not the same as appending a number to each of the terms (e.g. combining two infinite series as per my example).
But even if what you wrote were “correct” by the same token that the sum of the divergent series I mentioned is, it doesn’t have much to do my point in that paragraph, which was to say that these kind of statements make no intuitive sense but yet have some correctness to them.
It’s just a for loop.
There’s a smaller domain where we can validate the proposed counterexamples. To borrow from your example:
“this is obviously false” 27^5+84^5+110^5+133^5=144^5.
How is this a break down? (I don’t know what you’re building.)
That wording at the end suggests a typo.
It’s not a for loop, for loops don’t deal with infinity as far as I know.
As in, the results 1 + 2 + 3 + 4 … = −1/12 is “obviously false” yet mathematically true. So the pattern of “if something is true in a very intuitive way than it must be mathematically true”, doesn’t hold in those kind of cases (as opposed to the 2 * 3 = 6 case, where mathematics correctly describe what we intuit to be true by saying the statement is correct, at least if you think of mathematics as a “language” in the “programming but running on wetware” sense.
This only seems to be the case because the equals sign is redefined in that sentence.
I’d wouldn’t say it’s “true”.
Unless you think 1=0.
Proof:
[1] x = 1 + 1 + 1 + ….
Subtract 1 from both sides.
[2] x-1 = 1 + 1 + 1 + …
Substitute using [1].
[3] x-1 = x
Subtract x from both sides.
[4] −1 = 0
Multiply both sides by negative 1.
[5] 1 = 0
I’m pretty sure appending a single number to an infinite series is not the same as appending a number to each of the terms (e.g. combining two infinite series as per my example).
But even if what you wrote were “correct” by the same token that the sum of the divergent series I mentioned is, it doesn’t have much to do my point in that paragraph, which was to say that these kind of statements make no intuitive sense but yet have some correctness to them.
They are correct if you accept a strange premise like “infinity = 0” or ignore mistakes, like the one I made in the proof above.