An irrefutable demonstration of a conclusion, possibly via sequence of steps or combination of elements.
2. In what context is a "proof" embodied?
Do you mean what the range or domain of the proof are? Not sure on the "embodied". I think you mean the communication of the proof and the expected base knowledge to understand the proof.
3. Why is scribbling lines on a paper (that look like math to humans) 'more' of a proof than visual diagrams, if at all?
You seem to be focusing on the representation of the proof in a particular notation, rather than the actual logic of the proof.
The graphical demonstration leads to false conclusions. For example, if a=0, it implies that a^2 - b^2 is 0 (or it requires some unfamiliar graphical representation of negative areas)
4. If you came across a proof that was persuasive to alien intelligences -- and led them to conclude true things were true and false things were false -- but, alas, you did not understand it, does that make it less of a proof?
Again, the representation is not the proof, it is a means to record or communicate the proof.
If the representation implies that false things are true (e.g., if a==0), then it is not a proof.
- When I said "embodied" I roughly mean the ground rules that someone needs to know to check the proof. In the case of symbolic logic I mean the symbols and the transformation/rewrite rules. But I'm not sure yet how I would formalize the analogous concepts for visual proofs.
- Re: "You seem to be focusing on the representation of the proof in a particular notation, rather than the actual logic of the proof." ... yes, but maybe not necessarily. The "logic" of a visual proof is quite different than the "logic" of a symbolic proof.
We're on the same page, but if I were to take it one level deeper, I would add:
a. While I know what you mean by "irrefutable", I wouldn't use that word, because it sounds too much like "untestable". The whole idea of a proof is that each step can be verified. With a big enough lookup table, a proof can be checked in linear time (right?). If a step does not "obviously" follow then the step is not well-explained (in the "trivial to verify" sense).
b. Your choice of "demonstration" is key here. An essential aspect of a proof is that it is easier to check than generate. Simply "read off" each line of the proof and check against some known set of facts and transformation rules.
c. It is useful to distinguish between a verified proof (which is subject to correction!) and just a proof (which is a form, a way of communicating how something can be verified). See this Stack Overflow page: "Widely accepted mathematical results that were later shown to be wrong?" [1]
- irrefutable is "cannot be denied or disproven" - it means absolutely proven. Very different from untestable (which means you can't verify or prove it).
This is precisely the distinction we've been calling out from the original - it's a demonstration that works for some cases. The diagrams fail as a proof because they can be refuted by the negative/0 cases where they don't work.
- testing or verifying with a set of data is also very different from a proof. This "checking" or "demonstrating" provides some assurance of correctness or utility for the test domain.
- demonstrating and checking against known facts is not sufficient for a proof - "I've tested my division function for millions of positive and negative integers and real numbers! I even verify by multiplying the quotient by the divisor and I've proven it is correct!" did you happen to test a divisor of 0? (dividing by 0 can also invalidate proof attempts that do not exclude 0 as a divisor)
It needs to be proven to hold for all cases, not just a sampling of cases (though it is valid to define the range of a proof - the example could have stated, "this is a proof of the equation for positive values of a and b, and were b < a" maybe that could constitute a visual proof for that domain?)
An irrefutable demonstration of a conclusion, possibly via sequence of steps or combination of elements.
2. In what context is a "proof" embodied?
Do you mean what the range or domain of the proof are? Not sure on the "embodied". I think you mean the communication of the proof and the expected base knowledge to understand the proof.
3. Why is scribbling lines on a paper (that look like math to humans) 'more' of a proof than visual diagrams, if at all?
You seem to be focusing on the representation of the proof in a particular notation, rather than the actual logic of the proof.
The graphical demonstration leads to false conclusions. For example, if a=0, it implies that a^2 - b^2 is 0 (or it requires some unfamiliar graphical representation of negative areas)
4. If you came across a proof that was persuasive to alien intelligences -- and led them to conclude true things were true and false things were false -- but, alas, you did not understand it, does that make it less of a proof?
Again, the representation is not the proof, it is a means to record or communicate the proof.
If the representation implies that false things are true (e.g., if a==0), then it is not a proof.