3 ms·

Q: If I have a rectangle "a" wide and "b" high, how do I find the width "c" of another rectangle which has the same area but is "d" high? A: I'll be pedantic a

Q: If I have a rectangle "a" wide and "b" high, how do I find the width "c" of another rectangle which has the same area but is "d" high? A: I'll be pedantic and take it in baby steps: Given: area = height * width Given: area(c, d) = area(a, b) c * d = a * b Because equivalencies of equivalents are equivalent (c * d) * x = (a * b) * x Equivalencies multiplied by the same thing are equivalent Q: Wait, where did "x" come from? A: Just bear with me... x = 1 / d Because I say so (c * d) * (1 / d) = (a * b) * (1 / d) Substitution again ... Blah blah blah c = (a * b) / d Tadah! Q: What if "d" is zero? A: (Chuckle) Division by zero isn't defined. Q: Why not? A: Well, we want division to be defined as the inverse of multiplication. Q: Inverse? A: If x * y = z, and we know x and z, what is y? x * y = z => z / x = y. Q: So why can't we divide by zero? A: Because if x is zero then y could be anything. It's not so much that division by zero is evil as it is that multiplying by zero does not have a unique inverse, and disallowing multiplication by zero would be just too weird. Q: Why don't we just say that y has a whole slew of possible values? A: Because we don't. Look, this is simple arithmetic, not some weird group thing. You wanted one equivalent rectangle and I gave it to you. If d is zero there's no c that could be big enough to equal a * b. (Before you ask, "infinity" is not a number.) Well, I suppose you could pick a number if... Q: ... a or b is zero. I have no idea if a or b or d will be zero. Can I still use "c = a * b / d?" A: Don't you have to wash the car or something? [edit: consistent use of "c" and "d"]

n4r9 8y ago
That's not intuition, though. That's relying on previously established abstractions and set-theoretic frameworks.
- mgsouth 8y ago
  That's part of it. The core difficulties here are the definition of "inverse" and knowing when it's applicable to use the transformations. The former is difficult to pin down in a totally formal yet useful form. The latter will in practice depend upon intuition; no one is going to back-compute invariants ("d = ab/c" is always legitimate in the larger context of "given b = ac*d ...")
pavel_lishin 8y ago
> Q: What if "d" is zero? You can avoid chuckling by pointing out that this is a nonsense question. A rectangle which has a height of zero isn't a rectangle at all - it's a line. And lines don't have areas. (You can continue chuckling after someone asks why lines have no area, I guess.)
- mgsouth 8y ago
  I'm not aware of a formal definition of rectangle which excludes zero height. Sounds like an intuitive leap :)
Twisol 8y ago
The premiere dialectic in this area is Lakatos' "Proof and Refutations", which I highly recommend in general, and has plenty to say about the present topic. > Q: Wait, where did "x" come from? A proof from intuition is not the same as a formal proof with an insightful step. I wager that many formal proofs (in Coq, say) would spur such "Wait, what is that for?" questions. In this case, the selection of some arbitrary non-zero real "x" is a creative step, not an intuitive one. The missing proviso that "x" is non-zero is an oversight, but it's a "small error", and -- with some experiential intuition -- can be recovered. (EDIT: The arbitrary variable is quickly overridden with a specific definition, but this is not typical unless there's a good reason to conceptualize it this way. If its value is obvious, define it up front and discuss why that value is the correct one.) > What if "d" is zero? This is (very nearly) a refutation, in the sense of Lakatos. We reflect and decide that our basic proof idea works for rectangles of positive area, and in some sense zero-area rectangles aren't really the objects we wanted to talk about. So we include this as an assumption. The digression about division by zero, inverses, etc. is valuable, but missing the point that this is an algebraic representation of a geometric problem. There's a lot that could be said about whether geometry is intuitive or formal, but I think the issue of divisibility by zero is side-stepping most of the interesting stuff. ...anyway, this is quite a convoluted proof. We assume a priori that for any rectangle R, its "area" A(R) is well-defined and computable. We'll assume that the rectangles are axis-aligned with side-lengths R.x and R.y for each axis. We could have a separate discussion and proof about the invariance of area under a change of basis. Theorem: Given a rectangle R with non-zero side-lengths R.x and R.y, and given a non-zero real number c, there exists a rectangle R2 with non-zero side-lengths where R2.x = c and A(R) = A(R2). Proof: Because R2.x is already specified, we need only determine R2.y. We know that, for rectangles, A(R) = R.x * R.y. So we desire that R.x * R.y = R2.x * R2.y. Because R2.x = c and c is non-zero, we have the equivalent desire that (R.x * R.y) / c = R2.y. Thus, the rectangle where R2.x = c and R2.y = (R.x * R.y) / c will have the same area as R and will have the desired side-length c. [] English? Yes. But it has more similarity with legalese than informal English. Each clause makes direct appeal to some rule of logic, and it could be transformed into a Coq proof (or any other proof assistant) without much ceremony.
- mgsouth 8y ago
  > ...anyway, this is quite a convoluted proof. Agreed :) > Theorem: Given a rectangle R with non-zero side-lengths... And this is the key problem with the "proof" I laid out. My theorem does not specify non-zero dimensions. Therefore standard arithmetic division by a dimension is not well-defined. Since we're pondering intuition, it's interesting to consider how fixing the "proof" was approached. On the one hand, you fixed it by in effect back-progating invariants to the problem definition. Another approach would be to use a modified definition of "division" ("d = 0 if a or b are zero"--would still require the constraint "c !=0 unless a*b=0"). Mathematically, the former appoach is almost always what you're going to want, since a modified definition will be pretty useless to build upon consistently. From an application standpoint, the latter would be much more robust. This is kind of one of the points made in the Sci Am article--pure vs. applied proofs are different.

n4r9 8y ago

That's not intuition, though. That's relying on previously established abstractions and set-theoretic frameworks.

mgsouth 8y ago
That's part of it. The core difficulties here are the definition of "inverse" and knowing when it's applicable to use the transformations. The former is difficult to pin down in a totally formal yet useful form. The latter will in practice depend upon intuition; no one is going to back-compute invariants ("d = ab/c" is always legitimate in the larger context of "given b = ac*d ...")

pavel_lishin 8y ago

> Q: What if "d" is zero? You can avoid chuckling by pointing out that this is a nonsense question. A rectangle which has a height of zero isn't a rectangle at all - it's a line. And lines don't have areas. (You can continue chuckling after someone asks why lines have no area, I guess.)

mgsouth 8y ago
I'm not aware of a formal definition of rectangle which excludes zero height. Sounds like an intuitive leap :)

Twisol 8y ago

The premiere dialectic in this area is Lakatos' "Proof and Refutations", which I highly recommend in general, and has plenty to say about the present topic. > Q: Wait, where did "x" come from? A proof from intuition is not the same as a formal proof with an insightful step. I wager that many formal proofs (in Coq, say) would spur such "Wait, what is that for?" questions. In this case, the selection of some arbitrary non-zero real "x" is a creative step, not an intuitive one. The missing proviso that "x" is non-zero is an oversight, but it's a "small error", and -- with some experiential intuition -- can be recovered. (EDIT: The arbitrary variable is quickly overridden with a specific definition, but this is not typical unless there's a good reason to conceptualize it this way. If its value is obvious, define it up front and discuss why that value is the correct one.) > What if "d" is zero? This is (very nearly) a refutation, in the sense of Lakatos. We reflect and decide that our basic proof idea works for rectangles of positive area, and in some sense zero-area rectangles aren't really the objects we wanted to talk about. So we include this as an assumption. The digression about division by zero, inverses, etc. is valuable, but missing the point that this is an algebraic representation of a geometric problem. There's a lot that could be said about whether geometry is intuitive or formal, but I think the issue of divisibility by zero is side-stepping most of the interesting stuff. ...anyway, this is quite a convoluted proof. We assume a priori that for any rectangle R, its "area" A(R) is well-defined and computable. We'll assume that the rectangles are axis-aligned with side-lengths R.x and R.y for each axis. We could have a separate discussion and proof about the invariance of area under a change of basis. Theorem: Given a rectangle R with non-zero side-lengths R.x and R.y, and given a non-zero real number c, there exists a rectangle R2 with non-zero side-lengths where R2.x = c and A(R) = A(R2). Proof: Because R2.x is already specified, we need only determine R2.y. We know that, for rectangles, A(R) = R.x * R.y. So we desire that R.x * R.y = R2.x * R2.y. Because R2.x = c and c is non-zero, we have the equivalent desire that (R.x * R.y) / c = R2.y. Thus, the rectangle where R2.x = c and R2.y = (R.x * R.y) / c will have the same area as R and will have the desired side-length c. [] English? Yes. But it has more similarity with legalese than informal English. Each clause makes direct appeal to some rule of logic, and it could be transformed into a Coq proof (or any other proof assistant) without much ceremony.

mgsouth 8y ago
> ...anyway, this is quite a convoluted proof. Agreed :) > Theorem: Given a rectangle R with non-zero side-lengths... And this is the key problem with the "proof" I laid out. My theorem does not specify non-zero dimensions. Therefore standard arithmetic division by a dimension is not well-defined. Since we're pondering intuition, it's interesting to consider how fixing the "proof" was approached. On the one hand, you fixed it by in effect back-progating invariants to the problem definition. Another approach would be to use a modified definition of "division" ("d = 0 if a or b are zero"--would still require the constraint "c !=0 unless a*b=0"). Mathematically, the former appoach is almost always what you're going to want, since a modified definition will be pretty useless to build upon consistently. From an application standpoint, the latter would be much more robust. This is kind of one of the points made in the Sci Am article--pure vs. applied proofs are different.