5 ms·

This is best considered an illustration, rather than a proof, as it fails for any of the following: a=0 and b is non-zero b>a (result is negative. If you swa

This is best considered an illustration, rather than a proof, as it fails for any of the following: a=0 and b is non-zero b>a (result is negative. If you swap the variables, you have to imagine the area is a negative area) either variable is negative

dbsmith83 2y ago
I think you could make it work for negative numbers. You would just shade the negative area differently to show it. For example, if you subtract 1000 sq ft from 100 sq ft, geometrically it means you are trying to remove more area than you have. So, to represent this, draw a rectangle of 100 sq ft. Conceptually "extend" this area by subtracting 1000 sq ft. The extra 900 sq ft that you try to subtract but can't "exists" as a negative representation. This could be represented by flipping the surplus 900 sq ft into a negative axis or shading it differently to denote that it's a deficit rather than an actual, positive area.
xpe 2y ago
A koan in four parts: 1. What constitutes a proof? 2. In what context is a "proof" embodied? 3. Why is scribbling lines on a paper (that look like math to humans) 'more' of a proof than visual diagrams, if at all? 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?
- kbutler 2y ago
  1. What constitutes a proof? 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.
  - xpe 2y ago
    To answer some questions: - 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] [1]: https://mathoverflow.net/questions/35468/widely-accepted-mathematical-results-that-were-later-shown-to-be-wrong https://mathoverflow.net/questions/35468/widely-accepted-mat...
    - kbutler 2y ago
      Couple of points: - 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?)
Ma8ee 2y ago
> Result is negative. If you swap the variables, you have to imagine the area is a negative area And why is that a problem? Just reverse the equation before and after the proof.
- kbutler 2y ago
  It means it isn't a "proof", it's an illustration of a partial domain of the original equation.

dbsmith83 2y ago

I think you could make it work for negative numbers. You would just shade the negative area differently to show it. For example, if you subtract 1000 sq ft from 100 sq ft, geometrically it means you are trying to remove more area than you have. So, to represent this, draw a rectangle of 100 sq ft. Conceptually "extend" this area by subtracting 1000 sq ft. The extra 900 sq ft that you try to subtract but can't "exists" as a negative representation. This could be represented by flipping the surplus 900 sq ft into a negative axis or shading it differently to denote that it's a deficit rather than an actual, positive area.

xpe 2y ago

A koan in four parts: 1. What constitutes a proof? 2. In what context is a "proof" embodied? 3. Why is scribbling lines on a paper (that look like math to humans) 'more' of a proof than visual diagrams, if at all? 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?

kbutler 2y ago
1. What constitutes a proof? 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.
- xpe 2y ago
  To answer some questions: - 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] [1]: https://mathoverflow.net/questions/35468/widely-accepted-mathematical-results-that-were-later-shown-to-be-wrong https://mathoverflow.net/questions/35468/widely-accepted-mat...
  - kbutler 2y ago
    Couple of points: - 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?)

Ma8ee 2y ago

> Result is negative. If you swap the variables, you have to imagine the area is a negative area And why is that a problem? Just reverse the equation before and after the proof.

kbutler 2y ago
It means it isn't a "proof", it's an illustration of a partial domain of the original equation.