Division is a sort of measurement. How many magnitudes A do I count in magnitude B? Magnitude A is a measure of magnitude B.

Every measure contains in some way that which it measures.

Zero is not a magnitude but the absence of magnitude.

Division by 0 is therefore incoherent. How many non-magnitude magnitudes do I have in magnitude A?

Division by zero is indefinite, because a contradiction accepted can produce any conclusion.

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Interesting. I was always taught the answer of “any number divided by zero” is “infinity.” And this was presumed to do some heavy lifting in more complex mathematical proofs. I was also taught that numbers are ideas or abstractions, and not “magnitudes.” On this account it would seem that zero, as an idea, could be used properly in mathematical division. Sorry this is just a bit of me thinking through what you’ve written, and I realize it’s a bit tangential, but I’m wondering about a distinction between number as idea and number as magnitude.

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Wood:

I think you are confusing calculus limits with the value of the function.

Technically, the limit of 1/x is infinity as nonzero x approaches zero.

But at x=0, 1/x is undefined.

The limit tells you ‘where the function goes’ as x gets arbitrarily close to zero but – the important bit – does not actually equal zero.

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Zippy,

Very much thanks! That helps me think through this very much.

SA,

Thanks for the post BTW. I spent the whole afternoon thinking of the ontology of “zero” and “infinity.” I think I was getting annoyed at what I presumed was equating “undefined” or “indefinite” with “incoherent” and the ramifications that had for other things I’m partial to (particularly the notion of “infinity.”). But on the second thought I think I’m just misunderstanding. Anyway, thanks again for an interesting post.

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I’m glad that you found the post useful. As with most of my posts, it began by me Googling something stupid. On this occassion, I googled Siri jokes on my iPhone one of which was “what is zero divided by zero?” I leave the hilarity for you to discover.

I took the notion of indefinite or undefined to be along the lines of “could be anything,” which would be a result of an incoherent proposition rather than the same as incoherence.

It is particularly important that we know what we mean when we talk about “zero” or “infinity.” If by infinity we mean, as Zippy notes, as the divisor becomes arbitrarily small the resultant will be arbitrarily large, then things are peachy keen. If we mean, 1/0=∞, we’re just talking nonsense.

A related problem that I tried to explain to a physicist (originally discussing Aristotle’s doctrine of motion) was trying to talk about a line as if it were constituted by an infinity of points.

The more I think about my mathematics education, the more I realize the number of misleading terms and handing waving of incoherence.

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If by “at x=0, 1/x is undefined,” we mean “at x=0, 1/x is a meaningless proposition” then I’m ok with that. If we mean, as I was inclined to think, “1/x at x= 0 is undefined but meaningful in some way like it is just bigger than big,” then we’re in trouble.

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Wood:

You have to be careful about that sort of thing. It arguably drove Georg Cantor mad.

That is mostly tongue in cheek, but at the fringes there probably is real danger in treading too much where angels won’t. A disproportionate number of the very best mathematicians went insane.

I’m reminded of the movie

Event Horizon. A group of scientists attempt to make a spaceship which can travel faster than light, by generating a tiny black hole through which the ship can travel. They accidentally open up a portal to Hell. Literally.LikeLike

semioticanimal:

I think “undefined” is the algebraic (if you will) equivalent of “incoherent”. The formalism cannot tell you anything about what it does or doesn’t mean at x=0: x=0 literally doesn’t exist as a part of the formalism, so the formalism itself tells you nothing at all about how to

interpretx=0 metamathematically (that is, ‘outside’ of the formalism).Metamathematically taking the formalism itself as dispositive on some (any) value of the function at x=0 is self contradictory, since the structure of the formalism itself tells you that the value is formally undefined at that point. That’s what discontinuity

means: the function has no value at that point.In this particular example (as opposed to, say f(x) = 1/|x|) even the calculus makes this obvious, because Lim x-> 0+ is positive while Lim x->0- is negative. There is ‘infinite’ distance between the limits depending on the side from which x=0 is approached.

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An algebraic analogy for why liberalism is rationally incoherent: because ‘live and let live and do not discriminate’ is literally undefined at precisely the place where it has to be concretely well defined, the place where politics becomes

actual: in the authoritative resolution of controvertible cases through actual decisions by authority.LikeLike

It seems that discontinuity may itself be problematic, though I may be thinking of this in too geometric of terms.

A line is terminated by a point, which is the bound of its extension. In this case, the point of zero is both the terminus of the line and something the line cannot contain. I think talking about a stepwise function such as f(x) = 1 for x less than 1 and 2 for x greater than 1 is easier to visualize. x=1 bounds both lines, but neither contains it.

I suspect rather that algebra is a deeper abstraction than geometry, but I don’t know.

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Well, in the case of f(x)=1/x it isn’t a line at all, it is an asymptotic function, and the discontinuity isn’t a simple “hole in the line”.

Discontinuities are very commonplace. But by definition a function is undefined exactly at the point of discontinuity.

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I admit that I’m still working this out.

Put differently, f(x) is a continuous object that is unbounded at the point of discontinuity, that is the point it ceases to exist. By unbounded I do not mean that the range goes arbitrarily large, but there is no point in the domain that terminates the function from where it does not exist. This is described simply as a limit, which is equivalent to saying there is no bound.

What discontinuities in the relevant sense do you see as commonplace?

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Mathematical discontinuities like the one at x=0 for f(x)=1/x are very common. Physical discontinuities are pervasive, where one thing ends and another begins. In quantum physics many properties are discrete not continuous.

One way to think about discontinuity and incoherence would be to consider a dog before you and ask what color is the part of the dog that is on Mars. The answer is undefined because the dog isn’t on Mars, it is here: no part of it is on Mars. Any concrete answer to the question isn’t even wrong, because the question itself contradicts the reality it purports to be about.

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I think I see what you mean, but I would dispute the use of the term “discontinuities” used generally. Discontinuity includes the notion of a privation, that is something should be continuous but has ceased to be. The dog isn’t exactly discontinuous but finite, that is bounded. Discrete entities aren’t continuous to be begin with so calling them discontinuous in someway seems off and to confuse the manner of measurement or representation (i.e. on a continuous line) with as it is in itself.

The discontinuity of the function seems to be essentially different because it is precisely that it is not bounded on the domain. That is there is no x that is the end or bound of its existence, where x=0 cannot be this because the bound is contained in the continuous thing, but x=0 is precisely where it does not exist. This seems to be in the very nature of division and continuity in that it does not end and considering the function as in some manner actualized (even mentally) is to actualize an infinity which is in itself incoherent. So, the incoherence does not seem to be simply in the discontinuity, but in taking the function as some whole when it is in fact unbounded even excluding explicitly the discontinuity.

As with the physical incoherence, I’ll put it in my own abstract terms for my better understanding: “Given what is not, how are things about that?” Given that there is a king of France, what is his name? The question as conditioned on something contrary to reality excludes reality from the answer, but truth is a correspondence to reality and therefore can neither be true nor false, since it already excludes the measure by which to be judged.

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What would be the Thomistic interpretation of, say, the surreal numbers or the different levels of infinity?

For example, Cantor discovered that there were multiple levels of infinity, one ”countable” and the other ”uncountable” in size.

Mathematicians picked up from this and have discovered that there exist inaccessible cardinals, which is basically what happens when you take infinity and make the same conceptual jump you make when you go from 0 to infinity and apply this to infinity and see where you land. This has also been expanded into indescribable cardinals, which is yet another conceptual leap away; all the way up super huge and n-huge cardinals which are many more conceptual leaps away.

The conceptual leap here talked about being of the value that it brings you from zero to infinity, but applied to infinity as the starting point instead of 1, so as to travel beyond regular infinity.

The surreal numbers, on the other hand, are basically a mathematically ordered system where omega represents infinity, but arithmetical operations of omega minus 1 or plus 1 being both valid entities AND bigger than any finite quantity too.

It also includes infinitesimals which complicates things even more.

Now, what do Thomists make of all of this?

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I generally don’t honor myself with the title of “Thomist”, more from a lack of knowledge than desire or inclination. However, I recall Fr. Benedict Ashley, OP dedicating a chapter to mathematics in his book “The Way Toward Wisdom”.

What I make of it is that we’d need to set up the conceptual framework better first. Much of talk around infinity appears to be generally incoherent or at least in some respect problematic. Much of it, as I understand, revolves around set theory, which is itself extremely problematic, particularly when considering infinite sets where “set” denotes something whole and bounded and “infinite” denotes something unbounded. Fr. Benedict discusses this in his book. Therefore, I would begin by questioning “countable” and “uncountable” infinities to begin with.

Another mathematically inclined “Thomist” is Michael Flynn, who would likely have many valuable insights and is more knowledgeable about advanced mathematical concepts than I.

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