Discussion in 'Physics & Math' started by Seattle, Jun 24, 2018.
Feel free to leave whenever.
Log in or Sign up to hide all adverts.
We have an example in "motion pictures', where the mind melds a sequence of still photos into an apparent continuum.
Then there is your computer screen, where a sequence of on/off pixels, is interpreted as a moving thing.
My brain is having a hard time with the idea of motion being discrete.
Are we saying that what appears to be a moving mass might actually be a mass that is standing still, then instantaneously moving to another position where it is then standing still, then instantaneously moving to another position where it is then standing still, etc? Wouldn't that violate all sorts of laws of physics, accelerating and decelerating without cause, and exceeding the speed of light by moving instantaneously, etc?
And if the reason it does not violate all sorts of laws of physics is because the distance moved is infinitesimally small (limit approaching zero), or because the time the mass is sitting still is infinitesimally small (limit approaching zero), then aren't we back to continuous motion rather than discrete motion?
I'm referring explicitly to the equations of motion that work. Not intuition.
Of course. But so far we have failed to do that. The discontinuous models don't work. Hence my observation of positive evidence, a step beyond allowed possibility.
You think you can put together a mathematical model of motion that produces positions, speeds, curvatures, etc, that match the current models and current measurements in every respect down to Planck dimensions, and then varies from them.
I handed you an item of evidence - the accuracy of the current equations of motion.
Meanwhile, the fact that we can't measure exactly supports the hypothesis of actual continuity - if physically real, the fuzz covers the holes in the mathematical real line, in the language above. That would take care of some of the nagging problems mentioned above.
Yes it's possible for infinity to be more than mathematical. ANYTHING that remains forever is infinite; any argument or statement can be infinite, providing it is always true.
That's simply false. It's a factual falsehood. It's wrong. What's funny is that you already gave the precise definition of an infinitesimal, namely a positive number that's less than 1/n for any positive integer n. The concept of measure has nothing to do with this.
Note that in measure theory, the measure of any individual point or real number is zero. That goes for standard real numbers and infinitesimals alike.
The infinitesimals of the 17th century were replaced by the modern theory of limits. Cauchy, Weierstrass, Dedekind, et. al. You know this, right?
Secondly, actual infinitesimals were formalized within set theory by Hewitt in 1948 and Robinson in 1960-something. You know that too, right? You quoted that fact in your last post! And now you pretend to be unaware of it. It's a puzzler.
You know there's been a lot of progress in mathematics since the 1600's, don't you? It's like saying, "Well ever since the phlogiston theory, nobody knows what heat is." We've learned a lot since then.
But what does that have to do with the "common speech" meaning of measure?
Meta: The last few days when people quote me I don't get an alert. This just happened today with your two most recent posts in which you quoted me. I didn't find out till I clicked on the thread and looked. I posted a query in the Feedback area. Are you by any chance doing anything unusual when you quote me?
This happens to be a very interesting remark, even if only a link. I happen to have a literally pre-fabricated rant. The state of the Wikipedia page on infinitesimals is one of my pet peeves in life. I don't edit Wiki but I do comment on Talk pages once in a while. I have on my list of projects to post to the Talk page for this article a detailed and somewhat savage critique of the page; along with specific recommendations to improve it.
Just for starters they don't even give the actual mathematical definition of infinitesimal. And their discussion of the hyperreals is extremely naive and misleading. Some of the claims they make are flat out false. They don't put any of it into context. They link very misleading and disingenuous sources. I could go on. I can provide specifics.
It is not your fault that this particular page is exceptionally bad relative to other technically-oriented Wiki pages. But when you reference it, I'd appreciate your quoting the exact sentence or words that you find important to your argument, and point me to them; so that I can have an opportunity to put them into their proper context.
In short: The Wiki page on infinitesimals is awful and is not to be trusted or taken seriously. If you are interested in anything they say, feel free to run it by me for context.
Well arfa brane now you know how I feel about that!!!!
Well this is actually a very interesting point. What is our level of discourse here?
This is a math and physics board, with the primary emphasis on physics, as I've observed. So if we are talking about infinitesimals, the math conception and the physics conception are on the table. And perhaps the philosophical conception, since philosophers since Leibniz have speculated on the subject. Probably before Leibniz too.
But now you are proposing something different. You say we are talking about what the average person, going about their business in the street, would say if you aimed your smartphone at them and asked them what they think about infinitesimals.
Personally I wouldn't have high hopes for such an enterprise. As Winston Churchill once said, "The greatest argument against democracy is a five-minute conversation with the average voter." And when it comes to abstract technical and philosophical concepts like infinitesimals, it's even worse.
If I'm exaggerating your position, it's only for effect. But what meaning am I to ascribe to the phrase, "common speech?" In a conversation about physics and math I'm supposed to forget what I know about measure theory -- one of the core mathematical foundations of QM by the way -- and instead converse on the level of some member of the general public? Have you seen the state of the general public lately????
arfa brane, I wonder if you misspoke yourself here. You don't really mean to suggest that we should abandon all hope of speaking precisely and scientifically, do you?
When I took calculus, I recall having some conceptual problems when the lecturer introduced infinitesimals. I've since realised they didn't have to introduce them because we were doing limits anyway.
Nonetheless, the idea that you can make the sides of a right triangle as small as you like and preserve their ratio (and the angle between them), and this is the same (or a very similar) concept as a limit (viz Fermat's method, the method of exhaustion etc), didn't appear all that controversial.
Incidentally, if as you say the measure of a single point is zero, what's the measure of an infinite number of points in a straight line?
As to common speech, most people would probably understand that although objects have continuous motion, measurement of distances is never exact. Most people would understand a non-zero number which is too small to measure (in the same context as measuring a finite distance). I say this means it's too small to calculate or compute. But we have derivatives, dy/dx has a definite meaning, it isn't infinitesimal or uncomputable because the ratio and the angle between dy and dx (90°) are the same as for y/x.
As the Wiki page says:
, implying that if you have an infinitesimal which is not compared to another, you have a number which is too small to be computed.
And no, I don't think I should believe everything I read on Wikipedia. Articles published as course material by lecturers from recognised universities is probably more authorative.
But I notice that the concept of infinitesimal quantities goes back a long way, and it's only recently that mathematicians have defined them rigorously. Which means Newton and Leibniz didn't have rigorous definitions (I suppose they might have had "common speech" definitions instead).
Well, I wasn't referring to that. There are two very different aspects to that: the "on/off" aspect and the "moving thing" aspect..
What you're talking about are effects of the way our visual perception works. You could say it's a physical effect. The signal coming from the world is changing too fast and/or our visual system has too much inertia. You could argue that it's been selected so by evolution but only as a result of a trade off between efficiency and the cost of a more accurate system.
I was instead talking of the intuition we have about the continuity of movement. This requires a much more elaborate interpretation of the signals, to construct a model of the world in terms of objects, moving objects. I can only guess that this model is built by the brain from experience. The individual's own experience of his own physical environment, during his life, from birth. That's a very different thing from inertia in our visual system even if in the end it all comes down to evolution.
OK, not much else to say. I've said all that was needed. It's all there.
So i guess we disagree.
Your description is a first draft. You need to move to 2.0.
If space is discrete then movement between one position and the next is not somehow equivalent to the kind of movement we have intuitively. You need to think of it differently. Movement in a discrete space is more like a change in the value of the position. There would be nothing else to it. Just like a value on a chart, going up and down, without inertia, so to speak. The laws of physics as we know them would only apply to the macroscopic result, a notion we're already familiar with. That requires a total flip of the way we think of the material world, i.e. our intuition of it. But our intuitions of the physical world are operational at our level as biological organisms, and therefore not necessarily true of fundamental physics. So no inertia, no speed.
We necessarily would have to "be back" to apparent continuous motion in terms of the macroscopic effect as detected by our science. The scale of the discreteness of space would have to be small enough to be compatible with our best scientific model, but given the inaccuracy of our measures, there will always be the possibility of that.
Calculus class (as taught in the US) is not rigorous and is generally the cause of many misperceptions. My grad advisor, true story. very eminent mathematician, said to me once: "Freshman calculus is a futile exercise in mindfucking."
If you learned from calculus class that infinitesimals are finessed by limits, you had a good teacher and learned a lot more than the average person in that class.
You're right, in fact it's quite intuitive. Limits were intuitive to the ancients. It was only the question of how to properly formalize them . This is the classic example of physicists using "bad math" and then leaving the mathematicians to sort it out. Newton's fluxions, the "ghosts of departed quantities" as Berkeley famously snarked, took two full centuries to rigorize.
Of course the rigorization is not necessarily the reality. The map is not the territory. How the world "really" works, I don't know and neither does anyone else. Particularly the people who think they know.
Deep question and math has an unsatisfactory answer. Of course as you know. The measure, in this case the length, of the unit interval [0,1] is 1. And the length of the interval [0,2] is 2. We know from Cantor that these two intervals and all others like them have the same cardinality, since there is an obvious bijection between them. It's just a simple scaling factor.
So how on earth do we add up uncountably many zeros to get 1, or 2, or even infinity? The measure of the interval [0, +infinity) in the extended real number system is +infinity. Those symbols have nothing to do with philosophical or even the other mathematical infinities, they're just a shorthand for unboundedness.
The philosophical answer is: Nobody has the faintest idea (of how infinitely many zero-size points make up a nonzero sized length). Euclid didn't tell us, when he said two lines meet at a point. Nobody knows the answer.
In the mathematical discipline of measure theory, we get around this problem by requiring that a measure must be "countably additive." That means that if I have sets of measure 1/2, 1/4/ 1/8, 1/16, 1/32, ... respectively, and these sets are pairwise disjoint, then their union has measure 1/2 + 1/4 + 1/8 + ... = 1.
But an uncountable union does not necessarily have that property.
The thing is, systems with infinitesimals do NOT shed any light on this philosophical problem. In fact the problem gets worse. Because any model of the real numbers containing infinitesimals is necessarily topologically incomplete. That means there are "holes" in the real line. Cauchy sequences that do not converge. This is one of the many reasons that the hyperreals don't help anyone's argument about pretty much anything. If someone had a real use for them we'd hear about it.
That's funny. I can't even get most people around here to agree to that. Let alone going out on the street, startling someone out of their smartphone trance and asking them, "Say do you believe that physical measurement can ever be exact?" I imagine they'd just start filming me as some street weirdo.
arfa brane I'm really serious about this. I can't imagine why you are trying to reduce our conversation to arguing over what "average people" think about anything? People don't think about this stuff. We have a precise technical vocabulary for talking about continuity and infinitesimals. That's the only coin of the realm on a physics and math board. Why am I even having to explain this?
Your claim here is that "most people" would hold YOUR interpretation. I see. Well that certainly settles it. Have you taken a poll?
By "most people" do you mean a random person on the street? Are you aware that "calculating" and "computing" are technical terms that you are using in the wrong context? Get this: Most real numbers are not computable Haven't we been through this a couple of weeks ago in this thread? So you are simply wrong on the established technical facts of the matter.
But if you want to argue that most average people, who never think about math or science and communicate in tweets, would agree with whatever you say to them about this, I'll certainly stipulate to that.
It's often not the same for ANY y/x. That's the whole point. Back to calculus class for you. The sequence 1/n for positive integers n has the limit zero; but none of its terms are ever zero. That's the whole point of limits. dy/dx doesn't need to be ANY of the values delta-y/delta-x ever took.
A factual error on the page then. "Computed" means a real number that can be arbitrarily approximated by a Turing machine. Most real numbers are not computable. The concept simply has nothing to do with infinitesimals. The Wiki author is simply wrong. Not wrong simply on a fact; but in their entire conception of what computability is. That article is just awful.
That's true but I said something much stronger. I said that even given the general unreliability of Wiki, the page on infinitesimals is PARTICULARLY AWFUL. You just quoted me a howler I hadn't even seen before.
And now you're about to quote me another. You quoted me this presumably from the same execrable Wiki page:
To give it a meaning, it usually must be compared to another infinitesimal object in the same context (as in a derivative). Infinitely many infinitesimals are summed to produce an integral.
This is disgraceful. It's wrong. It's the fever dream of a C student who slept through calculus class and now thinks he understands the subject.
This quote no doubt refers to the notation \(\int_a^b f(x) \ dx\).
Perhaps we tell calculus students that dx is an "infinitesimal" increment in x. Perhaps some teachers do tell some students that. It's good enough for freshman calculus. It's not the right answer. The expression dx is a "differential form," an abstract gadget that makes these formalisms all work out. It's taught in differential geometry. Even a lot of math majors don't get to see it till grad school. So there's no harm in people walking around thinking dx is an infinitesimal, until they show up to make a philosophical point about the nature of the world. At that point, what they don't know confuses things.
So this quote on Wiki is simply wrong. Integrals are NOT summations of an uncountably infinite collection of infinitesimally-thin rectangles, even though that's sort of the right intuitive visualization. Because we don't really have a way to add up uncountably many zero-sized things. Not in standard math and not in nonstandard math. The definition of the Riemann integral has nothing do do with infinitesimals. We take the limit of Riemann sums, remember?
This quote about integrals is awful. It literally makes me sick to my stomach. It's wrong for a technical page to confirm the worst misunderstandings of freshmen.
Still not tracking this reduction to the opinion of some jive-ass hipster on the street.
By the way although I don't know much about Leibniz, I do know something of Newton's work in this area. He did NOT use "common speech" ideas about anything. He labored mightily all through his career to explain what fluxions were. He well understood that there was a logical gap and he worked hard, and without success, to fix it. It took mathematicians another 200 years to solve that problem. But Newton well understood it.
My understanding by the way is that Newton didn't use infinitesimals. His ideas were closer to the modern idea of limits. It was Leibniz who was the big infinitesimal guy.
double post sorry
If we throw Einstein's special relativity into the mix, then whatever it is you are referring to as "the scale of the discreteness of space" would be subject to length-contraction when measured from a relatively moving reference frame. That doesn't seem to bode well for the idea of there being a "scale of the discreteness of space".
Can "Infinity" ever be more than a mathematical abstraction?
Infinity has physical reality that is described math formula: T=OK.
The focus of discussion would be the connection, correspondence, relationship if you will, between various mathematical abstractions and whatever in some kind of "physical world" they are held to be abstractions of.
So getting the math right would be part of the issue. But only part.
The correspondences between whatever is suggested by the IPO term "more than an abstraction", the abstraction itself, and the larger mathematical context in which the abstraction is embedded, are not going to be part of the mathematics involved. There is not going to be a mathematical proof that any given mathematical abstraction is "the" correct and complete one, that any given model is the only or best - or that any given interpretation of the math is correct, only, or best.
In particular, one may legitimately suggest, without loss of care or sense or even rigor, that the real line with its noncomputable numbers subtracted models physical distance without holes, and with plenty of infinities, and capable of modeling continuous physical motion.
One analogy might be with our abstraction and modeling of distance via the Pythagorean theorem. There are always two square roots involved, mathematically - we simply exclude one of them. A distance is a positive number. Likewise, any product of cause or chance is a computable number - we can say.
Infinity has physical reality that is described by math formula: T=OK.
Which physical formulas can be used to understand the infinite
Zero vacuum structure : T=OK ?
At first we can use the laws , formulas, equations of ''Theory of Ideal Gas''
because ''ideal gas''has the same temperature: T=OK.
Perfectly well agreed.
Not sure what IPO means here. Initial Public Offering is the only interpretation I know but apparently you mean something else. Regardless, I can't find anything objectionable here.
Perfectly well agreed.
No, this could not possibly be. The set of computable numbers (that is, the reals minus the noncomputables) are topologically incomplete. Most Cauchy sequences fail to converge. This is a fatal problem for anyone claiming such a system models the intuitive idea of continuity. Constructive mathematicians do try to patch this flaw by saying that the only functions that exist are computable ones, but that just compounds the philosophical error IMO.
In fact this is EXACTLY why QM is done in Hilbert space, which is defined to be topologically complete. It's so that every sequence and series which "should" converge, does converge. That's why topological completeness is important. That's why we traditionally model continuous motion using the real numbers and NOT the rationals or the computable numbers. There aren't enough rationals and computables to allow the convergence of those sequences and series that morally "should" converge. Only the completion of the rationals, namely the reals, fits the bill.
Seriously irrelevant. But please do read up on constructive mathematics to understand some of the subtleties involved.
Separate names with a comma.