The point used in geometry is not rendered mystical by Zeno's paradox, at least in the above phrasing. It is mystical because it is undefined. It fails the basic test of getting everyone who hears it to understand the same thing. Most people who hear the word "point" imagine a dot, but then we are told the dot has no width. Well then now there is no longer anything to picture about what was ostensibly to be understood by visualization (left undefined precisely because of the appeal that "everyone knows w hat a point is from their own experience").
The end result is an all-purpose heuristic phantom that can be used to help fudge through proofs so that the illusion of rigor can be maintained.
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Space and time have points. There is a point in space and a point in time. They are not the same thing, of course. Matter is made up of atoms, which are not points.
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The end result is an all-purpose heuristic phantom that can be used to help fudge through proofs...
This is blatantly false. Which mathematical proof do you think is fudged through?
Every intelligent discussion of any topic whatever has to have undefined words. Proof: every language has a finite number of words. Given word A, it can only be defined in terms of the other words of the language. Say when defining word A I use word B. But what is the definition of B? It has to use either word A [thus creating a circular definition, which is no defintion at all], or use a third word, C. Continuing in this manner, going from C to D etc, we exhaust all the words in the language. So when you get to the last word in your language, word Z, you either have to leave it undefined, or else define it in terms of earlier words used, making your definitions circular, which is even worse than undefined [because it makes all the words in the circle undefined, and yet gives them the illusion of being defined].
All proofs involving points accept their being undefined, and merely use assumed properties of points, which are all laid out clearly in advance.
That misunderstands the point: if space is discrete...
Are you saying that the assumption of continuity of space and time means no resolution of Zeno's Paradox? That the only way out is to dump calculus and geometry?
Point: undefined. Mathematicians could define it, but to do so would destroy their whole system of geometry because they have to use it with different definitions in different cases. They must have their equivocable entities in order that their logical deductions and constructions can be made to appear rigorous.
Point in space: granting we have a solid definition of "point," it still would only be a point in mathematical space, not physical space (physical space is just nothingness).
Point in time: an instant? In the context of motion, there are no instants - motion is always motion over an interval. See Zeno's other paradox of the arrow.
So you think Math is one big con job. OK. We are too wide apart to be able to continue this fascinating discussion.
Zeno's paradox is also a mathematical paradox, independent of reality, and has to be resolved in that context as well. For example, if someone came up with a seemingly solid proof that 2+2=5, that would have to be addressed by mathematicians. It would not suffice to say that Plato is wrong and that numbers don't really exist and are really just nothingness.
Motion is indeed only over an interval, but have you never heard of instantaneous velocity, the velocity at a point? The whole point [excuse the pun] of calculus is to define and use the concept of velocity at a single point. Are you saying calculus is wrong?
In any case, Zeno's Paradox as stated by the OP does not involve motion except as being over an interval.
A point must have no width for most purposes, but must have width for points to be able to be used to construct lines and planes. Geometry is equivocations all the way down, and you're right: equivocation is an accepted part of mathematician culture.
Words can only be defined by more words? Sounds reasonable, until you realize that you can just point to stuff and name it in order to identify it. In fact this is the very appeal upon which the need to define terms in geometry is shirked in the first place.
It is hardly convincing to claim that a term is "atomic" and need not be defined when people don't even understand the same thing by the term in question. It is a rather odd position to maintain that defining the key terms of a theory at least to the point that readers are on the same page is unneeded.
The supposed rigor of calculus and geometry is indeed a farce, but this is obvious from the start in so much as they do not unambiguously define their terms. They arrive at the correct answers most of the time simply because fudges have been added on top of other fudges until finally they have reached some sort of workability in fields like engineering where there is a market test. The problem with this approach is that it makes these subjects far more difficult than they need to be, and the underlying lack of rigor while having the appearance of rigor is epistemically hazardous for other fields that try to incorporate mathematics (physics being the premier example).
"We are too far apart" in what? Opinion? Position? Hopefully matters of logic are not something one can have opinions about or take positions on. If they are, I side with Voltaire: "If you wish to converse with me, define your terms." Otherwise people just go around in circles until some convince everyone else to submit to the authority of their genius.
A point...must have width for points to be able to be used to construct lines and planes.
Maybe in bizzaro world. But in this world points never have width, even when used to construct lines and planes.
Geometry is equivocations all the way down...equivocation is an accepted part of mathematician culture.
Please show me one instance of equivocation in math or geometry. Hint: there are none.
So you think that the key terms of every theory are always defined? Quite the opposite is true. The key words are never defined, and cannot be defined, because the key words are, by definition, those words that will not be defined.
...you can just point to stuff and name it in order to identify it.
Really. How are you going to point at a point? Or a number? Or a homomorphism? Or a set? Or justice? Or truth? Or thinking? Or Aristotole [or any dead person]? Or a country? Or a marriage? Or love, hate, joy? Or human being, or human action, or means, or ends? You see where I'm going with this.
The whole point of abstract reasoning is to go beyond limited caveman ways of doing things to achieve greater understanding.
...the need to define terms in geometry is shirked...
In which discipline do you think the atomic terms are defined, whether by pointing to the stuff or otherwise?
Short answer: None. The study always begins with listing the undefined terms, and stating explicitly the properties these undefined terms are assumed to have. [For example, a point has no length, width or breadth]. There usually follows some informal attempt to help the reader get what is meant [perhaps something like "the center of a circle is an instance of a point", with a picture drawn], but the informal stuff is never used in the proofs.
I don't know where you are coming from with all this, AJ.
In any case, my last post on the subject. Someone else will have to defend geometry's honor.
Physiocrat: I really don't care whether something has been "universally accepted" for thousands of years. If it is false, it is false.
I really don't care whether something has been "universally accepted" for thousands of years. If it is false, it is false.
This is perfect.
Smiling Dave: Zeno's paradox is also a mathematical paradox, independent of reality, and has to be resolved in that context as well. For example, if someone came up with a seemingly solid proof that 2+2=5, that would have to be addressed by mathematicians. It would not suffice to say that Plato is wrong and that numbers don't really exist and are really just nothingness.
I agree that mathematics is a concise, deductive system and the paradox can be resolved within itself, but that would be in a vacuum. We are relating math to reality. The truth is that there is no such thing as numbers outside of what we write on paper and in our minds. Mathematics is a subjective system, and the truth is that while Zeno was attempting to point out that change is an illusion, he inadvertently discovered that math doesn't really exist, and neither do measurements. When I say they don't exist, I mean physically. There is no measurement outside of our perspective.
It's similar to the dollar bill, as Rothbard references in What Has the Government Done to Our Money? He says maleable currency is great for society because it can be fractionalized whereas if I wanted two eggs and only had one huge wheel barrow, I wouldn't get anywhere. In that case, 0.5 oz of gold certainly exists, but its fractionalized value is completely subjective. Example: ever tried to buy a chicken's eggs by offering the chicken 0.5 oz of gold? I'm willing to bet no transaction would happen because economics is a subjective construction that defines supply and demand and nothing more. We all simply know the rules to the game, so-to-speak, so we succeed at playing it. Zeno is simply inadvertently pointing out that it is all a game period.
Equivocation is inherent in the notion that a line is "made" of points that are themselves just collections of properties, or in the relation of such constructs to anything physical - depending on how exactly the math is phrased. Instantaneous velocity is another oxymoron born out of equivocation. Which basic term is being equivocated on depends on the actual definitions used, but I am suggesting that if you look for it, it will be there in every case (if you can ever get solid definitions out of a mathematician). But I bet you've never looked, perhaps because you trust that scientists cannot err about something so basic and agreed-upon. That view does not have a very good track record, though.
"Point to" as in show someone which aspect of their experience is being referred to by a word, and do this unambiguously enough that you can be reasonably sure everyone understands the same thing by the word, or close enough for the current context (but then you must be very careful when changing contexts). This may be hard to do, but that's a fact of life: communication is hard. Shirking the responsibility to communicate a theory clearly is shirking science and mathematics all together. However, my point here is not to deplore those who engage in this fanciful practice, but to point out that this is neither rigorous nor efficient.
Praxeology as expounded by Mises had solid definitions at its core, because readers can generally understand which aspect of their experience is being referred to (pointed to) by words like "felt uneasiness" and "pain". Just as the tribal elders would explain that diseases are caused by gods beyond all human experience, mathematics also purports to reference the unexperienceable, such no-size points, instantaneous velocities, and infinitely thin lines.
Staying home and praying to the gods works sometimes, provided the appropriate rituals are followed that accidentally help cure the disease. Similarly, math gets away with these unexperienceable entities only because because of the notion that these things must be intuitively experienceable by those higher-up mathematician demigods, and because this witchcraft ends up working by eventually stumbling upon the right set of fudges to patch over the basic slipperiness underneath. Naturally, a whole mess of paradoxes are left in the wake of these circumlocutions.
If even a social science like economics can be made rigorous, at least at its core, why give math a free pass? It is after all quite possible to visualize dots, lines, and numbers (as groups of dots, for instance). These are all quite experienceable and unambiguous in the relevant ways.
Smiling Dave:Autolykos, It might make sense, but it is not the direction humanity has taken ever since Euclid. I think it behooves us to find an answer to Zeno's Paradox consistent with a few millenia of the universal thinking of the mathematical and scientific community.
It might make sense, but it is not the direction humanity has taken ever since Euclid. I think it behooves us to find an answer to Zeno's Paradox consistent with a few millenia of the universal thinking of the mathematical and scientific community.
Dave, with all due respect, I really don't understand why you're being so vague here. It almost seems to me like what you're really trying to do is insinuate that you have a vastly greater amount of knowledge here than me, and that you think it's in my best interest for me to shut up and go away rather than presumably embarrass myself. However, I won't embarrass myself, because no matter how much knowledge you throw at me, I won't feel embarrassed. So please, go ahead and make your position crystal-clear.
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Autolykos: Smiling Dave:Autolykos, It might make sense, but it is not the direction humanity has taken ever since Euclid. I think it behooves us to find an answer to Zeno's Paradox consistent with a few millenia of the universal thinking of the mathematical and scientific community. Dave, with all due respect, I really don't understand why you're being so vague here. It almost seems to me like what you're really trying to do is insinuate that you have a vastly greater amount of knowledge here than me, and that you think it's in my best interest for me to shut up and go away rather than presumably embarrass myself. However, I won't embarrass myself, because no matter how much knowledge you throw at me, I won't feel embarrassed. So please, go ahead and make your position crystal-clear.
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