It is unusual to say that difference proves theory to be wrong. — Ludwig V
I would be happy to say, I think, that Zeno's application of the theoretical possibility of convergent series to time and space and the application in Thompson's lamp is a mistake. — Ludwig V
But calculus does have uses in applied mathematics, doesn't it? — Ludwig V
Non-dimensional points which have a dimensional separation? H'm. — Ludwig V
But then a boundary (between your property and your neighbour's) doesn't occupy any space, even though it has a location in the world and will consist of non-dimensional points. — Ludwig V
If there is a maximum number of divisions, then what is that maximum number? — TonesInDeepFreeze
Einstein’s General Theory of Relativity describes the properties of gravity and assumes that space is a smooth, continuous fabric. Yet quantum theory suggests that space should be grainy at the smallest scales, like sand on a beach.
One of the great concerns of modern physics is to marry these two concepts into a single theory of quantum gravity.
Now, Integral has placed stringent new limits on the size of these quantum ‘grains’ in space, showing them to be much smaller than some quantum gravity ideas would suggest.
...
Some theories suggest that the quantum nature of space should manifest itself at the ‘Planck scale’: the minuscule 10-35 of a metre, where a millimetre is 10-3 m.
However, Integral’s observations are about 10,000 times more accurate than any previous and show that any quantum graininess must be at a level of 10-48 m or smaller.
My question was about mathematics not physics. — TonesInDeepFreeze
Yes. The trouble is that the inapplicability of convergent series in certain situations does not, for my money invalid them in all situations.Why do you say this? Doesn't science proceed through the falsification of theories? — Metaphysician Undercover
Well, it would be interesting to know what your criterion of truth is in mathematics, if a calculation procedure is effective and useful.I don't deny that calculus is extremely useful, but that usefulness may be misleading relative to the goal of truth. — Metaphysician Undercover
I agree with everything you say.This is the way I understand boundaries between two pieces of private property. — Metaphysician Undercover
Thank you. That's very clear.If the discussion is about points in ordinary real 2-space or real 3-space then points are distinguished by being a different ordered tuple.
In 2-space, the point <x y> differs from the point <z v> iff (x not= z or y not= v).
In 3-space, the point <x y t> differs from the point <z v s) iff (x not= z or y not= v or t not= s).
If a particular line, say the ordinary horizontal axis, then <0 x> differs from <0 z> iff x not=z. — TonesInDeepFreeze
I'm puzzled. I thought you thought that Thompson's paradox was flawed and therefore invalid - as Thompson did, didn't he?There is no smallest number, but if paradoxes like Zeno's and Thomson's are valid then there is a smallest unit of space and/or time. — Michael
That seems to be the result of some recent research. But I don't think it applies to mathematics as such, and perhaps one ought to wait and see whether anything else emerges from research.I understand the view that there is no smallest number but that there are smallest distances and durations. But I am asking about the views of others too. — TonesInDeepFreeze
I'm puzzled. I thought you thought that Thompson's paradox was flawed and therefore invalid - as Thompson did, didn't he? — Ludwig V
What does "paradox is valid" mean? Does it mean that the premises indeed entail a contradiction. — TonesInDeepFreeze
The analogy with imaginary numbers and apples is amiss in this regard: Yes, apples are counted by integers, not imaginary numbers, so indeed imaginary numbers are not the correct kind of number to count with. But distances and durations are measured by real numbers, so smaller and smaller real numbers are not a difference in the kind of number. — TonesInDeepFreeze
But I am asking whether some people here do believe there is a smallest number. — TonesInDeepFreeze
Thompson says that there's a false premise, which is that infinitely many tasks cannot be completed in finite time. He says that there's no finite upper limit to the number of tasks that can be completed in finite time, but that not infinitely many can be completed in finite time. — TonesInDeepFreeze
Yes, that's what I thought. I think the concept of a valid paradox is a bit confusing.No, I think (as did he) that it successfully shows that supertasks are not possible. — Michael
But space or time being infinitely divisible does not entail that supertasks are possible.Yes. If space and/or time being infinitely divisible entails that supertasks are possible, and if supertasks being possible entails a contradiction, then it is proven that space and/or time are not infinitely divisible. — Michael
That's the first I've heard of any use of transfinite numbers in this thread. I don't think they are relevant - more, I very much hope they are not relevant.In this case the mistake is in the application of transfinite numbers. — Michael
How is that possible? Infinite means without limit.He says that there's no finite upper limit to the number of tasks that can be completed in finite time, but that not infinitely many can be completed in finite time. — TonesInDeepFreeze
I'm sure it could count as a human right. But can we also stand up for the right to form the inverse of any natural number? (For clarity, forming 1/2 from 2, 1/3 from 3 and so on. (I'm not sure whether 0 or 1 need to be included here.)I am a tough customer when it comes to giving up my natural prerogative to add 1 to any number. — TonesInDeepFreeze
That's the first I've heard of any use of transfinite numbers in this thread. I don't think they are relevant - more, I very much hope they are not relevant. — Ludwig V
But space or time being infinitely divisible does not entail that supertasks are possible. — Ludwig V
Yes, that's what I thought. I think the concept of a valid paradox is a bit confusing. — Ludwig V
Then, must mathematics not allow smaller numbers? — TonesInDeepFreeze
So, if someone claims that the mathematics is to blame, then we would ask whether the mathematics itself (which holds that there is no smallest number) needs to be rejected, or whether the way in which the mathematics is applied needs to be rejected, or both. — TonesInDeepFreeze
I've been thinking about this. My comment on this was wrong. Of course, one cannot complete infinitely many tasks in a finite time. "Complete" does not apply to infinite series, by definition.He says that there's no finite upper limit to the number of tasks that can be completed in finite time, but that not infinitely many can be completed in finite time — TonesInDeepFreeze
It depends on how you choose to analyse it.I'm not talking about physical possibility. But even then, if space and time are infinitely divisible then motion is a physically possible supertask. — Michael
If space and/or time being infinitely divisible entails that supertasks are possible, and if supertasks being possible entails a contradiction, then it is proven that space and/or time are not infinitely divisible. — Michael
The trouble with Thompson's lamp is that no switch can function in an infinitely small time. — Ludwig V
I'm sure it could count as a human right. — Ludwig V
the mistake is in the application of transfinite numbers — Michael
It's not that complicated. Imaginary numbers have a use – even in electrical engineering – but I cannot have an imaginary number of apples in my fridge.
There's nothing wrong with maths, just sometimes an improper use of it. There is no smallest number, but if paradoxes like Zeno's and Thomson's are valid then it would suggest that there is a smallest unit of space and/or time – and that this isn't just a contingent fact about the physics of our world but something far more necessary. — Michael
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