Hi everyone, welcome to another entry of our Short Attention Span Reading Group
The Text
We will study On Contradiction by Mao.
It is divided into 6 sections (7 if we count the very short conclusion), none of them will take you more than 20min to read (most will take less) :).
I think this essay can be summarized by its first sentence
The law of contradiction in things, that is, the law of the unity of opposites, is the basic law of materialist dialectics.
And this is all it studies, starting to what is the difference between dialectics and metaphysics, the law of contradiction, what are contradictions, how are they defined, what are their different types, and so on. And of course what it means for Marxism.
The biggest question I am left with after reading this essay is the place of Nature in materialist dialectics...
Supplementary material
- On Practice by Mao Tse-tung. It is significantly shorter than On Contradiction, and they both go hand in hand.
How is motion itself a contradiction? What things contradict each other?
I haven't read the relevant Engels work but I imagine he's invoking Aristotle's arrow paradox.
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The denial of a unique instantaneous position for a given time t flies in the face of pretty much all of kinematics.
You literally could not write a displacement function (because multiple positions at time t indicates the relation is not a function), and thus couldn't calculate velocity or acceleration as the derivative of that displacement function.
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That's true, but then I'm left to wonder how we're supposed to judge it's usefulness. If you're describing a model for the nature of motion, what are you after if not empirical adequacy?
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The linked essay has another example of a supposed impossibility, though the proof needs a bit of filling in:
It's correct that there are no real numbers that are the square root of negative real numbers. It's a real proof by contradiction if it's filled out properly.
How would it have been if they'd said to Descartes that complex numbers flew in the face of pretty much all of arithmetic? Because I know that complex numbers had opposition after they were invented.
A function can in fact take a set of positions (or some other type of thing containing multiple positions) and produce a new set of positions. Perhaps it's merely a useful simplification to say that there's a unique position.
We know that we can never find out exactly where a particle is due to the uncertainty principle and we know that we can't distinguish things that are separated by less than a Planck length, so is it possible even in principle to empirically determine for sure whether or not particles have a single instantaneous position?
It's not though.
Sure, but you're changing what he is saying. Saying that the "square root of negative integer is a real number" is a contradiction. Saying "the square root of -1 is an imaginary number" is not a contradiction, it's a stipulative definition of i.
Functions by definition produce a single unique output for a given input. To say that something is in two places at one is to say that you can't describe motion as a function of time, which is the opposite result of kinematics. See the second image in the linked section.
That's not what Heisenberg's uncertainty relation says. And wouldn't matter anyway because epistemic uncertainty is different than ontological non-existence. Not knowing the exact position doesn't not proscribe the existence of an exact position.
Engels isn't a mathematician. To demand that he get correct every detail of a proof he may have heard once and that he has no training to reproduce is a bit overblown when the conclusion is correct and the true proof that he's obviously referencing would have been produced historically.
This is blatantly incorrect. Function outputs don't have to be unique. That's an injective function. If functions produce one output, that one output doesn't have to be a single number, it can be a single set of positions. You can have a function that takes in the starting set of positions and produces the resulting set of positions.
You must be having an argument with someone else in your head, because my point was that we can't know for sure whether or not particles have single positions, not that single positions are proscribed.
You're confused
Function outputs must be unique for a given input. At time t, the position function must have a exactly one output. That can be a vector if you like, but it must always be the same vector for the same given t.
You're confusing this with one-to-one, which would be to say the position at every time must be different than every other time. Which is not what I'm talking about, because I've always been focusing on some given time t
Yeah, well your buddy Engels here knows they can't, because of dialectical gibberish. And you're still using a popscience misrepresentation of the inequality. The position of a quantum particle can be known to an arbitrary high position (i.e, as exactly as you want). That just has consequences for our knowledge of the momentum.
You are apparently unaware that sets can contain sets, so the element of set Y in your definition can itself be a set, so the output of a function can be a set.
The position of an arbitrary particle can't empirically be known as exactly as you want because there are limitations to measuring devices, as you are apparently also unaware.
Nope, like I said, it can be a vector, or a set, if you like. But it's always going to be the same unique set for a given input
Which has also nothing to do with the uncertainty principle; it's independent of any sort of technological limitations.
I agree. I earlier thought by 'unique', you meant unique in the sense that no other input produced the same output, but I see that I was mistaken. Since I see what you mean, I don't know why you said it, because I never implied that a function could give different outputs for the same input. The same output set would always be produced for the same input set in what I said.
Now that we're agreed that a function can output a set, a set of positions at a point in time can be transformed by a function into a set of positions at a later time, meaning that it's theoretically possible for kinematics of a more complicated variety to handle the motion of particles with multiple simultaneous positions.
As far as the uncertainty principle, I may or may not be wrong about it, but my main point stands: empirically, we don't know whether moving particles have unique instantaneous positions or not because we can't measure to the exactness needed to determine that. Theoretically, this seems to be the case as well, which is why I mentioned the Planck length.
One possible alternative would be that it could be that the particle occupies all the positions of a too-small-to-measure segment along the direction of travel, for example. I'm not saying that this is the case. I'm merely trying to give the benefit of the doubt to Engels. I don't want to summarily dismiss his work just because it doesn't meet my preconceptions of how kinematics work.
Of course it's possible to describe all motion in such a formalism, but why?
Imagine a particle moving in R^1 at 1 unit/second that occupies, at time t=0, the position x = 0. How would you meaningfully write the motion of this particle as a multi-valued function, and what is the benefit of doing so, apart from shoehorning in Engels' conclusion?
It's not that it doesn't meet ones preconceptions; It can be made compatible with any physical/mathematical theory, and any physical mathematical theory can be made compatible with it. But that's true of essentially any axiomatic assertion on the nature of things, so I don't know what makes the law of contradiction/the dialectic particularly meaningfully.
As far as how it could be done in that hypothetical world: you move each position according to how far the velocity says it would move and you return the set of results.
[Edited because this webpage is wildly closing the editing field and either deleting or submitting the contents of it]
As far as why? Generality for the theory. No one cares what kinematics says about the movement of particular dust particles on a particular exoplanet, but it's nice to know that kinematics works generally. The same would be true for dialectics.
The point of dialectics is what can be predicted usefully from it, I think. I'm still new to it, so I'm still waiting to see what that is. The thing about motion isn't that useful, it's more a thing about making the theory general, but there should be results that are useful.
As far as why I personally proposed that formalism, it was because you claimed that multiple positions implied a non-function relation, which isn't necessarily the case.
I mean it does; can you write a set of ordered pairs describing the motion of the particle above at certain points in time that
And if so, how and why?
Sure.
As for how, the first element of the ordered pair is a set of starting positions. The second element of the ordered pair is a set of ending positions. ({start_0, ...}, {end_0, ...}). The function is, of course, a set of these ordered pairs where each ordered pair's first element is unique in the set.
The X in your definition of function is the same set as Y: the set of sets of positions.
As for why, just to demonstrate that the statement was incorrect.
No, I mean actually do it for the above problem set up. With the actual numbers.
You mean to write the infinite set of ordered pairs of infinite sets? No, I can't quite do that, as it would take infinite time.
Not all of them, just a few of them. I think I know the solution you're couching in the abstract terms above, and I want you to explicitly lay it out so we can look at how absurd it is.
Let's say at t = 1, t = 1.5, and t = 3.
It doesn't quite matter how absurd it appears to you. What matters is that it fulfills the definition of function you said it didn't.
Sure it matters. I've already acknowledged you can shoe-horn the assertion into any system. But I've also pointed out that this makes the assertion meaningless.
So now I'm looking to see if you can provide me a kinematic example of a particle moving in R1 occupying two places at once, where the second point it's occupying isn't meaningless nonsense.
I don't care what you acknowledge. I can see that a function can take a set of positions and produce a set of positions, and that's good enough for me.
Times.
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