Quick Links are within the YouTube video description itself... details are below:
A variety of individuals briefly describe what they think infinity really means.
Explains about the prominent philosophers in history that have tried to understand the perplexing nature of infinity.
Explains various philosophical ideas which associate mathematics with religion.
Aristotle's realisation that actual infinities cannot exist in reality, led him to invent the term: “potentially infinite” (Misnomer; see website definitions), i.e., progressing indefinitely.
Talks about Georg Cantor, and explains the significance of his role in relation to infinity and mathematics. Mentions Cantor's mental illness and depression, his stints in sanatoria, and the criticisms he suffered from other mathematicians.
Georg Cantor: 1845 - 1918
Henri Poincaré: 1823 - 1891
Leopold Kronecker: 1854 - 1912
See: 15:45 onward to hear the following quotes:
“So, the big insight from Cantor, the biggest of all, is that there are different kinds of infinity – some infinities can actually be bigger than others...”
“He was not regarded, necessarily, as a hero to all mathematicians. In fact, one of the greatest mathematicians of his era, Henri Poincaré, described Cantor’s work as a disease – Ouch.”
“Another one of his colleagues, a mathematician named Kronecker, who was in a position of power in Berlin... Kronecker described Cantor as, not only a charlatan, but a corrupter of youth. Now I don’t know what that makes me, since I see there are some young people here, and I will be talking about his work.”
Cantor's work could have been described more politely of course... maybe a nice letter... something like:
“Dear Georg, please think carefully about what you mean by different size limitless states of existence, and until you can provide a coherent explanation as to how pre-existing limitless state (i.e., actually already exists) can somehow exist in different specific sizes in order for those sizes to be different to one another... then please don't teach such beliefs to anybody, particularly not to young students that might simply accept what you’re telling them.
By all means proceed in teaching people all about the meaningful relationships that you've discovered, but please use more appropriate terminology instead of simply asserting the existence of infinity. And be very cautious about mentioning platonic numbers, and if you do... don't forget to mention that they must still exist as specific numbers in the platonic realm.”
Yours sincerely, Leopold Kronecker... or Henri Poincaré.
Using 3 dots “...” at the end of a row which represents a finite number of guests, does not transform the number of guests to become infinite, i.e., no longer existing as a numerical value – the same sin is committed on behalf of the buses.
Adding numerically to any row of guests, or to any column of buses, indefinitely, is the only way to increase their size – any set’s size will always be finite in reality.
Follow this link to see an alternative explanation: Hilbert’s Hotel – Presented by: Professor Raymond Flood
Changing values down the diagonal of any finite list will indeed produce a new unique number.
However, the list will have simply increased in size by 1, as explained in more detail below:
Any sequence (set) of real numbers will never actually exist to be infinite in size – adding to any such list indefinitely will result in: either reaching the set's limit if it is finite, or continuing indefinitely until the adding process is no longer possible...
Examples: our universe progresses to become another universe via a big bang and nothing survives the transition... or you simply run out of particles.
The diagonal argument’s purpose as presented in the main video, is to prove (by producing a new number) that an assumed finite list of real numbers cannot actually be finite, i.e., fit inside a container, e.g., belong to a completed list or set.
Slightly irrelevant but amusing... notice that the above “purpose” is a contradiction in itself – any new number that is produced must have already existed in the list, simply because the assumed list was indeed a completed list – in other words, when a new number is discovered, why wasn't it already in the list? ...
Even the list itself is paradoxical... to produce the result (the new number) would mean having to use one of the digits in the new number, simply because the new number should be in the list to begin with – the inverse of the: “grandfather paradox” via time travel... i.e., the new number cannot be added to the list unless it exists 1st... vs... you cannot be in the past to kill your grandfather if you were never born.
By assuming there exists an abstract infinite list of natural numbers, and an abstract infinite list of real numbers, is to assert that both have an actual limitless state of existence – therefore it is ridiculous to then assert that one of those limitless states is larger than the other – don’t believe it!
Obviously, by analysing how one column of single digits corresponds to one column of multiple digits, the opportunity arises to create additional entries to the multiple digits column – and the diagonal number method does indeed guarantee that a new unique number will be produced…
Example:
1 --- 0.1
2 --- 0.22
3 --- 0.333
4 --- 0.4444
Example rule could be +3… which produces the new number: 0.4567
This correspondence analysis simply applies to a finite number of rows – to then assert that numbers exist platonically and therefore different size infinities exist, can lead one to the realisation that platonic numbers do not actually exist.
In summary via the above link... the definition of a numerical value is just that – a specific value.
Also follow this link: Infinite Sets – Different Size Infinities – Seriously?
Therefore, a platonic “specific value” must still be finite – and because the very nature of the idea of platonic existence must be a limitless one, platonic numbers simply cannot exist.
This additional short video below, corresponds very nicely to: 25:05 within the main video at the beginning of this page.
Note: although it is indeed a great explanation about correspondence, the term “infinity” is simply inappropriate, i.e., new terminology is needed for any such analysis – assuming that numbers exist platonically is where all the confusion seems to originate from, although maybe not for this gentleman, hence “the claim is, there exists, somewhere somehow”... simply using the term “infinity” for convenience perhaps.
Four simple alternative ways to increase the size of such a list are:
Add a new number which is larger in value than the largest existing number in the list – examples: largest number + 1... or × 2.
Add a new number which is smaller than the smallest existing number – examples: smallest number - 1... or - 2.
New smallest number between 0 & 1: simply insert a 0 anywhere between the decimal point and the last digit of the smallest existing number.
New largest number between 0 & 1: simply insert a 9 anywhere between the decimal point and the last digit of the largest existing number.
In other words, Georg Cantor’s diagonal argument is unnecessary for proving that lists can be increased in size – any list of real numbers can be increased in size very simply, and indefinitely... but it will always remain finite.
1st Mistake: suggesting it might be possible to create a list that contains all the real numbers.
Explanation: if such a list could exist it would be finite (hence “all”) – adding to the list would simply make it larger – not infinite.
2nd Mistake: not distinguishing between numbers that exist in reality and supposed platonical existence.
Explanation: the reason it is not possible to create a list that contains (or represents) a larger number of things than there are particles in the entire universe, is because real numbers do not have an independent (platonical) existence (see above links in previous section), i.e., the quantity of represented numbers in any list is simply limited by whatever the chosen container is – e.g., bits stored in a computer... or particles in spacetime.
[Quote: from Wiki] In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets.
It states: There is no set whose cardinality is strictly between that of the integers and the real numbers. [/Quote]
It is impossible to identify size differences when comparing different infinities, because infinity is not a size (logically speaking) – it's an idea – a misconception.
No doubt some incredibly significant, interesting, and useful analysis has occurred – it's just not actually about infinity – it would help if terms were used other than “infinity” and the like, accompanied with language that describes the purpose without asserting that actual infinites exist, i.e., by using appropriate descriptions where applicable... not implying infinity.
Hugh Woodin states that infinity might just be “... human imagination gone wild... it really could just all be a fiction... we could go the sceptics route, and show once and for all, that the conception of mathematical infinity is just that, a fiction, and there is no answer...”
A general explanation describing why infinities are an issue in physics.
Raphael Bousso explains that when physicists encounter infinity, they know something is profoundly wrong; it signals there’s a crisis – the discovery of quantum mechanics a little over 100 years ago was a result of such a crisis.
A calculation that an infinite amount of light energy is emitted from an ordinary lamp, led Max Planck to desperately come up with a fix... energy can only be emitted in finite packets of energy – quantum mechanics had been born.
Raphael Bousso: “In this universe, there are infinitely many places where you win the lottery, and of course infinitely many places where you don't – and, on what basis then, do we predict, that you probably aren't going to win the lottery... we need to get rid, of these infinities...”
Philip Clayton: “There are at least three different subjects being talked about… a mathematical challenge (... actually describes it as a purely mathematical challenge) ... then we have the physical challenge… then we have these broader philosophical or metaphysical questions, that preoccupied the thinkers that I described...”
“... we’ve got three different spheres... could it be no contact between them?”
Hugh Woodin: “Any axiomatic system is subject to the possibility of being refuted. Now, if the axioms for number theory were shown to be contradictory, that would be a crisis, because that would wipe out all of mathematics; it would wipe out the foundations of physics.”
Raphael Bousso: “... really there’s always an incredibly small chance that you just got lucky with your experiments, and actually the theory is wrong...”
“... quantum mechanics fundamentally doesn’t make sense, unless you are at least in principle, able to repeat an experiment infinitely many times...”
“... you cannot even in principle repeat an experiment infinitely many times. I don’t know whether we should read that as a sign that maybe quantum mechanics is after all, only some effective theory; some approximation to a richer structure that we have yet to discover...”
Philosophising about the idea that mathematics and physics might exist as fundamentally separate things.
Considering infinity to be simple & beautiful, which results in a nicer mathematics, even if it means that we just pretend it exists.
Steven Strogatz: “... whether infinity is real or not, like from a very practical point of view, it’s helpful to pretend that infinity does exist… calculus is based on the domestication of infinity...”
Raphael Bousso: “Infinity is a very good approximation in many cases, to large finite numbers… there’s no perfect circle in nature… there’s no number of infinitely many digits that we could actually write down… we certainly don't have an infinite amount of information in the universe, what we do with that is a separate question...”
Raphael Bousso explains how gravity as theorised in Einstein’s general relativity, breaks down at incredibly small distances; a point (singularity) – space becomes infinitely strongly curved; density becomes infinite in a finite amount of time, which physicists know is nonsense… hence, the search for a theory of quantum gravity is required – a theory of everything that would explain both quantum size existence and the very large.
Hugh Woodin explains that if the infinity that is named after him is falsified, he’ll resign his position, and that he puts his academic life behind it – all in good humour of course... lol.
He also predicts that infinity won’t be refuted within the next 10,000 years, and therefore: “how does one account for that, unless it’s a reflection of something that is meaningful.”
Steven Strogatz describes how his kids are intrigued by their family “Pi plate” – the real number Pi (π), which in principle represents an endless number of digits (precision) – seems to be directly connected to our reality.
“... there’s nothing more orderly as a circle, which Pi embodies – is the very symbol of order and perfection. And so, this co-existence of perfect predictability and order, with this tantalising mystery of infinite enigma, built into the same object, to me is part of the pleasure of our subject; of mathematics, and I suppose of infinity itself.”
Philip Clayton expresses the difficulty of “grappling with the complexity of the question... to the very limits of what the human mind is capable of”. Expresses his own very deep philosophical insight into how religion, mathematics, and physics, all share common ground in the world we live in today.
Carl Sagan: Astronomer and author of the book “Pale Blue Dot: A Vision of the Human Future in Space”
“Carl Sagan made this point in pale blue dot… satisfied with a notion of God so simple, and non-complex, not stretching us in the way that Steve’s answer just did, and you think wait – there’s gotta be something wrong with that – so I call for a philosopher, or a theologian, or a religiousness, or a spirituality… that would fight for that sense of something… far more deeply interfused; that’s worthy of an animal; of a species, that can grapple with the continuum hypothesis; that can grapple with finites and infinites in physics; that can think of its own ultimate fate, or whatever that ground of all things is – with not less complexity, and not less profundity, and not less open-endedness, than we’ve seen from my three colleagues to my left on this stage. If human beings could do that, maybe it wouldn’t be shameful, to include philosophy alongside mathematics and physics, in discussions like this.”