The pedantic mathematician in me though didnât like the concept in the âIntermediate Value Theoremâ section being described with that name. Iâm not sure that is actually the theorem being relied on here.
A couple of reasons:
IVT only applies to functions from the real numbers to the real numbers, whereas the example youâre using it on is a mapping from a discrete set to a discrete set (or at a stretch, from rationals to rationals). IVT doesnât apply there (e.g. the function x â x^2 on the rationals between x=0 and x=2 goes from 0 to 4 without ever passing through 2).
I donât think the IVT is âtrivially easyâ. Itâs a theorem that the formal definition of continuity aligns with our intuition (which is not trivial), and really it is only true because the real numbers have been specifically constructed to make it true (it is not true over the rational numbers), and the construction of the real numbers is very weird and non-trivial!
The actual concept youâre relying on in that section is actually a lot simpler I think. Itâs essentially just the âparadox of the heapâ. As you say, if the value of N things is V, and the value of N+M things is W, then the sum of the changes of each addition to N must add to (W-V). You donât need calculus for that. Itâs not the Intermediate Value Theorem. It really is just âtrivially easyâ I think. Iâm not sure if that theorem has a name.
Ironically I might also be guilty of using some technical terminology incorrectly here!
I had in mind the discussion on valuing actions with imperceptible effects from the âFive Mistakes in Moral Mathematicsâ chapter in Reasons+Persons (relevant to all the examples mentioned in the IVT section of this post), where if I remember right Parfit makes an explicit comparison with the âparadox of the heapâ (I think this is where I first came across the term).
It feels the same in that for both cases we have a function from natural numbers (number of grains of sand in our potential heap, or number of people voting/âbuying meat) to some other set (boolean âheapâ vs ânot heapâ, or winner of election, or number of animals harmed). And the point is that mathematically, this function must at some point change with the addition of a single +1 to the input, or it can never change at all. Moreover, the sum of the expected value of lots of potential additions must equal the expected value of all of them being applied together, so that if the collective has a large effect, the individual effects canât be smaller, on average, than the collective effect divided by the number of consituents.
I suppose the point is that this paradox is non-trivial and possibly unsolved when the output is fuzzy (like whether some grains of sand are a heap or not) but trivially true when the output is precise or quantitative (like who wins an election or how many animals are harmed)?
The relatively more orthodox view amongst philosophers about the heap case is roughly there is a kind of ambiguous region of successive ns where it is neither true nor false that n grains make a heap. This is a very, very technical literature though, so possibly that characterization isnât quite right. None of the solutions are exactly great though, and some experts do think there is an exact ânâ where some grains become a heap.
Also, pretty much every empirical outcome is potentially fuzzy in some possible situation. (Elections can be tied, whether an experience is painful and hence harm or neutral can be unclear etc.)
Maybe, although an election being tied is about the only way that particular example can be fuzzy, and there is a well defined process for what happens in that situation (like flipping a coin). There is ultimately only one winner, and it is possible for a single vote to make the difference.
Whether an experience is painful or not is extremely unclear, but if your metric is just something like ânumber of animals killed for meat each yearâ then again that is something well defined and precise, and it must in principle be possible to change it with an individual purchase.
It can also be indeterminate over a short time who the winner of an election is because the deciding vote is being cast and plausibly there is at least some very short duration of time where it is indeterminate whether the process of that vote being cast is finished yet. It can be indeterminate how many animals were killed for food if one animal was killed for multiple reasons of which âto eatâ was one reason but not the major one. Etc. etc.
Thanks! I agree the math isnât exactly right. The point about x^2 on the rationals is especially sharp.
The problem with calling it âthe paradox of the heapâ is to make it sound like an actual paradox, instead of a trivially easy connection re:tipping points. I wish I had a better terminology/âphrase for the connection I want to make.
I like this post!
The pedantic mathematician in me though didnât like the concept in the âIntermediate Value Theoremâ section being described with that name. Iâm not sure that is actually the theorem being relied on here.
A couple of reasons:
IVT only applies to functions from the real numbers to the real numbers, whereas the example youâre using it on is a mapping from a discrete set to a discrete set (or at a stretch, from rationals to rationals). IVT doesnât apply there (e.g. the function x â x^2 on the rationals between x=0 and x=2 goes from 0 to 4 without ever passing through 2).
I donât think the IVT is âtrivially easyâ. Itâs a theorem that the formal definition of continuity aligns with our intuition (which is not trivial), and really it is only true because the real numbers have been specifically constructed to make it true (it is not true over the rational numbers), and the construction of the real numbers is very weird and non-trivial!
The actual concept youâre relying on in that section is actually a lot simpler I think. Itâs essentially just the âparadox of the heapâ. As you say, if the value of N things is V, and the value of N+M things is W, then the sum of the changes of each addition to N must add to (W-V). You donât need calculus for that. Itâs not the Intermediate Value Theorem. It really is just âtrivially easyâ I think. Iâm not sure if that theorem has a name.
Whatâs the connection to the paradox? The sorites is far from trivial or solved.
Ironically I might also be guilty of using some technical terminology incorrectly here!
I had in mind the discussion on valuing actions with imperceptible effects from the âFive Mistakes in Moral Mathematicsâ chapter in Reasons+Persons (relevant to all the examples mentioned in the IVT section of this post), where if I remember right Parfit makes an explicit comparison with the âparadox of the heapâ (I think this is where I first came across the term).
It feels the same in that for both cases we have a function from natural numbers (number of grains of sand in our potential heap, or number of people voting/âbuying meat) to some other set (boolean âheapâ vs ânot heapâ, or winner of election, or number of animals harmed). And the point is that mathematically, this function must at some point change with the addition of a single +1 to the input, or it can never change at all. Moreover, the sum of the expected value of lots of potential additions must equal the expected value of all of them being applied together, so that if the collective has a large effect, the individual effects canât be smaller, on average, than the collective effect divided by the number of consituents.
I suppose the point is that this paradox is non-trivial and possibly unsolved when the output is fuzzy (like whether some grains of sand are a heap or not) but trivially true when the output is precise or quantitative (like who wins an election or how many animals are harmed)?
Thanks, I get what you meant now.
The relatively more orthodox view amongst philosophers about the heap case is roughly there is a kind of ambiguous region of successive ns where it is neither true nor false that n grains make a heap. This is a very, very technical literature though, so possibly that characterization isnât quite right. None of the solutions are exactly great though, and some experts do think there is an exact ânâ where some grains become a heap.
Also, pretty much every empirical outcome is potentially fuzzy in some possible situation. (Elections can be tied, whether an experience is painful and hence harm or neutral can be unclear etc.)
Maybe, although an election being tied is about the only way that particular example can be fuzzy, and there is a well defined process for what happens in that situation (like flipping a coin). There is ultimately only one winner, and it is possible for a single vote to make the difference.
Whether an experience is painful or not is extremely unclear, but if your metric is just something like ânumber of animals killed for meat each yearâ then again that is something well defined and precise, and it must in principle be possible to change it with an individual purchase.
It can also be indeterminate over a short time who the winner of an election is because the deciding vote is being cast and plausibly there is at least some very short duration of time where it is indeterminate whether the process of that vote being cast is finished yet. It can be indeterminate how many animals were killed for food if one animal was killed for multiple reasons of which âto eatâ was one reason but not the major one. Etc. etc.
Thanks! I agree the math isnât exactly right. The point about x^2 on the rationals is especially sharp.
The problem with calling it âthe paradox of the heapâ is to make it sound like an actual paradox, instead of a trivially easy connection re:tipping points. I wish I had a better terminology/âphrase for the connection I want to make.