1) IMO, the axiom of infinity is nonsensical and leads to absurdities.
2) Accepting the non-numeric / purely imaginary / unrealisable status of infinity implies:
- The commonly given definition of infinity is wrong: 'a number greater than any assignable quantity or countable number'
- Transfinite arithmetic is a work of pure fiction
- Ideas about space and time that assume the existence of actual infinity are not mathematically sound.
3) BTW Did you know the reason actual infinity is enshrined within maths as a number is that Cantor was a devout Lutheran.
4) My interest in maths stops when maths stops telling me about the nature of reality.
5) Even if infinity existed, finity is a subset of infinity and what works for infinity needs to work for finity also. One-to-one correspondence does not work for both so it is flawed.
6) If something goes on forever, you can't count it - even with an infinity of time it is not possible to measure something that goes on forever. This is what Galileo recognised and what Cantor ignored - and it leads to spurious results, such as the number of naturals is the same as the number of rationals - how can anyone swallow that? For each natural, there is an infinite number of rationals... One-to-one correspondence gives nonsense results.
7) In short (and my opinion only) - a number has zero width,
8) I find the concept of a dimensionless object difficult - it has no extents so it cannot have any existence - how can any sound reasoning performed with a non-existent object - assuming its existence (in order to reason with it) leads straight to a contradiction?
9) I will humor you. The number of points in [0,1] is uncountably infinite.
— softwhere
Measure theory does not seems to provide any justification for the above claim - neither do I see any justification anywhere else in maths.
10)For example, take the extended complex numbers - the set of complex numbers plus ∞. The definition used for ∞ is z/0=∞. Now you can call that ‘an assumption’ if you like (and a pseudo-justification in terms of limits can be given) but it is plainly a wrong assumption. I believe there are then fields of maths (like complex analysis) which build on the idea of the extended complex numbers. Then people in the physical sciences build further theories based on these ideas. The net result is whole vertical slices of human ‘knowledge’ which are based on wrong assumptions and are therefore not valid knowledge.
Similar bad assumptions to the above example can be found in the hyperreal numbers and the projectively extended real line. Another example, already discussed above, is the axiom of infinity from set theory - the assumption of the existence of actually infinite sets of objects. It is a bad assumption to make and set theory is based on that bad assumption. Many things in maths and science are then built upon the foundation of set theory. Again we have whole swaths of knowledge based on bad assumptions - all that ‘knowledge’ is therefore not valid.
I am not a mathematician italics added.
11) But assumptions that are plain wrong/bad (counter logical) lead nowhere useful, lead other folks (in the physical sciences) astray, and result in lots of clever folk wasting huge amounts of time on wild goose chases (eg a good portion of modern cosmology is like this IMO).
12) A 0-dimensional point can in no way be the constituent of a 1-dimensional line segment - the point has zero length and the line segment has non-zero length. So it is incorrect to say, as mathematicians often do, that a line segment contains an actual infinity of points.
Likewise, a 1-dimensional line cannot be the constituent of a 2-dimensional plain - the line has length but zero width so it cannot be the 'parts' of a plain (which has non-zero length and non-zero width).
13) The problem I see is that (applied) mathematics forms the basis for our understanding of reality. So scientists pick up definitions and theories from maths and apply them to the physical sciences.
Now the set of natural number exists purely in our minds - my believe is there is nothing in reality akin to it. So there is this impossible concept which is taken from maths and is being applied in the physical sciences - producing erroneous results - cosmology is the biggest offender.
14) I am not disputing it is possible to measure intervals, I am disputing the common mathematical claim that there is an actually infinity of points on a line segment length 1.
How many points do you claim there are on a line segment length 1? The answer must logically be one of the following:
1. Infinite number
2. Finite number
3. Undefined
(there are no other possibilities)
If it is [1], that means 1/0=∞ which is nonsense
If it is [2], then a point must have non-zero length which is not the definition used in maths.
So I contend it must be [3].
15) Bijection/one-to-one correspondence is a procedure that produces paradoxes like Galileo's Paradox, or the cardinality of the naturals is the same as the cardinality of the rationals. It is therefore to my mind an unsound procedure. Cantor did nothing to help our understanding of infinity IMO; he has lead us down the wrong path entirely. — "Devans99
As with the teaching of infinity, something which is just an assumption is taught to us as absolute knowledge. I feel our maths teachers are letting us down — Devans99
Now the prevailing wisdom is that [1] holds - a line segment is composed of an infinite number of zero length points. I cannot make any sense out of this. How can anything zero length (dimensionless) be said to composed something with non-zero length? This is the view Aristotle held. — Devans99
Nevertheless, the point I am making is that I can find no workable mathematical description of continua. This might lend credence to the idea that, like matter, time and space are discrete? — Devans99
If I say of myself that it is only from my own case that I know what the word "pain" means - must I not say the same of other people too? And how can I generalize the one case so irresponsibly?
Now someone tells me that he knows what pain is only from his own case! --Suppose everyone had a box with something in it: we call it a "beetle". No one can look into anyone else's box, and everyone says he knows what a beetle is only by looking at his beetle. --Here it would be quite possible for everyone to have something different in his box. One might even imagine such a thing constantly changing. --But suppose the word "beetle" had a use in these people's language? --If so it would not be used as the name of a thing. The thing in the box has no place in the language-game at all; not even as a something: for the box might even be empty. --No, one can 'divide through' by the thing in the box; it cancels out, whatever it is.
That is to say: if we construe the grammar of the expression of sensation on the model of 'object and designation' the object drops out of consideration as irrelevant. — Wittgenstein
If by workable you mean conformity to your private intuition of the continuum, then actual mathematicians have famously wrestled with this. https://plato.stanford.edu/entries/weyl/ — "softwhere
As far as I can tell from your posts, you think that math is some strange form of metaphysics that uses symbols as abbreviations for fuzzy concepts. And then proofs are just fuzzy arguments to be interpreted like mystical literature on the profundities of time, space, matter. — "softwhere
You're failing to consider the length that corresponds to each point in a line. So, although points are dimensionless, the distance between points have a dimension viz. length.
Considered another way there are an infinite number of points in any given line but the line is constituted of the distances between these points and not the points themselves — TheMadFool
I suppose you can view a line segment as constituted of points or sub-segments. Whichever way though, the length of the constituents has to be non-zero. — Devans99
I'd like to continue the discussion if you don't mind because I see what you mean but I feel, given that mathematicians don't make a fuss about points being zero-dimensional, you're in error. — TheMadFool
The line AB can be infinitely divided into infinitesimally small non-zero lengths and each length will always have a point associated with it. — TheMadFool
Aristotle made a fuss about zero-dimensional points being the components of lines so I feel the question can be regarded as an open philosophical question. — Devans99
But this (questionable) maths leads to the conclusion that all continua have the same length - both a segments and its sub-segments are continua so they would both have the length 1. — Devans99
What were Aristotle's objections to points being zero-dimensional? — TheMadFool
Are you saying that because there are an infinity of points in any given line that all lines have to be of the same length? — TheMadFool
1. Points have zero dimension
2. A continua has an uncountably infinite number of points
3. All continua have the same structure and cardinality
4. Therefore it follows that all continua have the same length — Devans99
As you will notice for every point on AB there will always be a unique point on CD i.e. the cardinality of the set of points on AB = cardinality of the set of points on CD. They're both infinite. — TheMadFool
However, notice that a point on AB has a different numerical value to the corresponding point on CD. They are different quantities and so add up to different, not same, lengths. — TheMadFool
I'm saying that your objections are more than a century out of date. — quickly
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