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“For the past decade he [Wildberger] has been working on a new, infinity-free of trigonometry and Euclidian geometry.” He is working to counter the fact that angles are related, via circles, to pi, with its endless digits following the decimal point. I lack the maths to evaluate Wildberger’s work, but Doron Zeilberger say of it: “Everything is made completely rational. It’s a beautiful approach.”

The "endless digits" business is irrelevant to infinity.

the definition of potential must be expanded to "work done to bring an additional unit charge from infinity".

I'm not sure if my lack of maths is an advantage or a drawback when thinking about infinity. In most scientific discussion it is a serious disadvantage, but perhaps infinity is sufficiently divorced from science to make a difference. For example: Woodin is a mathematician and he obviously sees infinity as part of mathematics, whereas he is quite prepared to remove infinity from physics. My own feeling is that mathematics in general, and set theory in particular, may cope very well with “infinities”, but they are ultimately finite. Physical reality, on the other hand, must have its basis in infinity, or there would be no physical reality. When Woodin says “It may well be that physics is completely finite”. I would agree, as long as he is talking about our finitely bounded understanding of physics.

Let’s see if I have this right.1/r is the reciprocal of r.If 1/r = 0, r = infinity.In order to work with this rather inconvenient infinity, those who work with practical reality perform a sort of “renormalisation” and leave infinity to the theorists.

Infinite series consist of infinite numbers of elements but can sum to finite values.

If I remember rightly infinity on a camera’s distance scale originally represented anything greater than one hundred times the focal length of the lens, and may still be used in that sense on occasions. Obviously this is only loosely connected to anything that might be considered genuinely infinite, and probably is not what the NS article suggests may be an illusion.

Quote from: Bill S on 26/08/2013 21:52:26If I remember rightly infinity on a camera’s distance scale originally represented anything greater than one hundred times the focal length of the lens, and may still be used in that sense on occasions. Obviously this is only loosely connected to anything that might be considered genuinely infinite, and probably is not what the NS article suggests may be an illusion. 100f makes sense for a terrestrial camera, where your horizon is a few miles and your resolution is limited by camera shake, but not for an astronomical telescope. Fortunately, classical geometry comes to our aid: "rays from infinity are parallel" so we can build and test a scope on the ground before putting it into orbit. Except of course for Hubble MkI!

I have no quarrel with infinite sets/series as mathematical tools. As far as the extremely small is concerned, I am fine with “infinitesimally small”. I wish scientists would use it instead of “infinitely small”, which, in my opinion, is tantamount to saying it cannot be further divided, even in principle. If space is continuous, this must be the same as saying it is nonexistent.

Unfortunately, there seems to be no term, equivalent to “infinitesimal”, to cover things that are “sort of” infinitely large.

I would certainly not “jump into the no-infinity boat”. Without infinity, what would I argue about? Seriously, though, I see no realistic way round the idea that something must be eternal/infinite, otherwise we would not be here. This has caused me to do a lot of thinking about infinity, which might well be considered as being philosophy rather than science. I would not argue with that, except to say that if it is something so fundamental to our existence, then perhaps it has as much right to a place in scientific thought as does the underlying “reality” of QM.

I don't read too many pop-sci books, but I wouldn't be surprised if they abuse terminology by saying "infinitely small."

Do you mean "big enough that we can treat it as infinity in our equations"? When we simply call it infinity and acknowledge that we mean "big enough that we can treat it as infinity."

Yep. And this is the type of infinity that many physicists (and probably all engineers) use. It's a nice shorthand for "very big."

So where is the common source for converging rays?

Quote from: JP on 28/08/2013 13:37:48Yep. And this is the type of infinity that many physicists (and probably all engineers) use. It's a nice shorthand for "very big." No. Infinity isn't shorthand for"very big" but for "bigger than anything you can define".Which makes an interesting point. I mentioned parallel rays of light as having their common source at infinity. So where is the common source for converging rays?

Quote from: JP I don't read too many pop-sci books, but I wouldn't be surprised if they abuse terminology by saying "infinitely small." Hunting through pop-sci books for examples would be very time consuming and of little ultimate value, but Paul Davies, John Gribbin , Peter Cattermole and Stuart Clark come readily to mind.Quote Do you mean "big enough that we can treat it as infinity in our equations"? When we simply call it infinity and acknowledge that we mean "big enough that we can treat it as infinity." That’s fair enough, and I how I now accept it when I read “infinite/infinity” in the context of science or maths. I wish the authors of pop-sci books would make this clear, as there must be a risk that their readers will conclude that they are talking about some sort of “absolute” infinity, when they are talking about an approximation. It is easy for “hitch-hikers”, like myself, to believe we have something sussed, just because an expert has said it. As a child, someone said to me: “Question everything, not just because you think the person saying it is wrong, but because you may have misunderstood it. Never believe you understand something until you know you understand it”. I still try to maintain that attitude, but sometimes it annoys the hell out of people!

....the idea of an electron orbiting an atom without spiraling into the nucleus....

In some cases such as the size of the universe, some theorists take it literally. This is where I hedge my bets since you'd have to come up with a testable hypothesis to distinguish between an infinite universe or not. As far as I know the question isn't settled.

Over 40 years ago I had a long discussion with a maths teacher about infinity. It culminated in his conceding that the series of whole numbers, although apparently unbounded in both directions, was not an example of true infinity.

Imagine my elation when I saw, in the New Scientist an article suggesting that some physicists were trying to remove infinity from scientific – and even mathematical – calculations.

This must raise the question: How can we test for infinity? Surely such a test would require an infinite amount of information.

Great, but does that mean that the electron/wave is stationary, or is that too intuitive for thinking about QM?

Do you still believe this? If so, why? It certainly isn't true. Infinity quite litteraly means without bound. In this case for the sequence (not series) of whole numbers, by which I assume that you're referring to a sequence of numbers which increases without bound, then it's an infinite sequence. I don't understand how anyone could conclude otherwise. Please explain.

Please post the reference to this article. I'd like to read it. I assume you've read it. Would that be a correct assumption?

That depends highly on what it is that we're trying to measure as being infinite. Can you give an illustrative example?

The key phrase in your question was "think of". By no means the same as "is".

Quote from: Bill SImagine my elation when I saw, in the New Scientist an article suggesting that some physicists were trying to remove infinity from scientific – and even mathematical – calculations. Please post the reference to this article. I'd like to read it...

If we regard the electron as being in any, or all, of an "infinite" range of positions at the same time, do we regard the electron as being in motion, or stationary?

Turfing the lawn is an interesting question. If you rotate a hyperbola around the y axis you will generate a solid with a finite volume but infinite area, so a finite amount of water will flood a hyperbolic lawn, but you can never buy enough turf to cover it.

Disregarding "regard" for the time being, the wave function does not go to zero anywhere, so there is an infinitesimal probability that whatever it describes could turn up anywhere at all. So in principle, however large you make your search radius, it could be outside.

If you rotate a hyperbola around the y axis you will generate a solid with a finite volume but infinite area

Quote from: alancalverdIf you rotate a hyperbola around the y axis you will generate a solid with a finite volume but infinite areaI could do with an explanation, here, please.

Quote from: JP In some cases such as the size of the universe, some theorists take it literally. This is where I hedge my bets since you'd have to come up with a testable hypothesis to distinguish between an infinite universe or not. As far as I know the question isn't settled. I run into a problem here. If the Universe started at the BB, how could it be infinite?I have been assured that it could have been infinite from the start. However, I have yet to find an explanation as to how something infinitesimally small could be infinite.

Bill, what are you saying was infinitesimally small? the universe or the observable universe?

We may be slipping into a terminological pitfall, here. What do cosmologists believe began its existence at the BB; the universe or the observable universe?

You may remember my reply to you here.. Reply #24.