The laws of physics are mathematical in their nature. Theories in physics are man-made descriptions that sometimes get superseded, modified, or refined. Likewise, mathematics is a subject developed by people, but unlike our physical theories, it is easy to consider maths as something which is exact in its nature (obviously this excludes ideas such as infinity and different size infinities, e.g., infinite sets or containers with things in them, such as numbers).
Therefore, it seems reasonable to consider that the exact behaviour of physical laws in our universe (not our approximate man-made theories) might be operating exactly mathematically, i.e., it is only the physical theories that we have developed that are imprecise at any point in history. However, it doesn't necessarily follow that all mathematical structures represent some physical existence or possibility in our reality, and in fact as Roger Penrose has pointed out, it is only a small area of mathematics that we use in science.
Platonic existence is mentioned from 7:10 onward.
If mathematics is a foundation which our physical theories are based on, then it seems reasonable to consider mathematics as really existing in the same way we think of physical laws existing in reality.
In this sense mathematics is something we have discovered.
Entertaining mathematical philosophy from: 6:20 to 9:30.
Further entertainment (albeit off-topic) – some ideas about consciousness from: 12:20 to 14:00... sending humans into space and budget concerns from: 33:12 to 35:30.
Obviously mathematical objects don't have any kind of independent existence, other than some of them existing as laws of nature (so to speak).
It’s not as if numbers or equations actually exist as some kind of abstract independent things or objects (platonical) in our reality.
In this sense mathematics is something we have invented.
Hypothetically, even if numbers are considered to have an actual independent existence, what exactly would a supposed infinitely large numerical value be?
It cannot be a numerical value because by definition that would make it finite – i.e., even when considering platonical existence, it's not possible for an infinite numerical value to exist – likewise, the same impossibility applies to supposed infinitesimal values.
Just as nonsensical, is the fact that any supposed infinitely large numerical value, must have values higher than itself – exactly when does a sequence of values transcend into becoming non-numerical? whether thinking platonically or not.
The following link refers to an intriguing post, within which it proves: Finite / 0, is not equal to Infinity
Nevertheless, any supposed result other than undefined is nonsensical – infinity is not even a numerical value, and yet infinity is sometimes the supposed result of a seemingly numerical calculation (existence) of reality.
Ironically, if we treat “Division by zero error” to = Infinity = something gone wrong = paradoxical – then it actually makes sense – Brian Greene has said before:
Quote: “if infinity turns up as the answer to something that we could directly measure with a piece of equipment, then we know, that our calculation must be wrong... if you find infinity, popping out of your equations for something you can measure, you better go back, think about those equations, modify them in someway...”
At least for infinitesimal values – it follows that any supposed measurable value is considered “wrong”, i.e., more convincing logic that confirms infinity is a misconception.
Because every finite value is logically infinitesimal from the perspective of an infinite value – then it follows that infinite values must also be “wrong”, or else all finite values would be physically paradoxical in contrast to infinitely large existences.
What will happen to the discovered (calculated) value of Pi?
Answer: Our computers will carry on calculating Pi (π) indefinitely, until they reach the limit of their capabilities, or they stop working, or maybe time itself will eventually stop – the number of decimal places calculated will never be infinite, obviously.
Quora link: Pi is not infinite - it is an “irrational number” – one of the methods for calculating Pi is the: “Nilakantha seriesr”
In contrast – “rational” number example: 1/3 * 21 = 7, or 21/3, and although 1/3 = 0.3333333 recurring, it’s not an actualised infinity of decimal places – the mathematical law (so to speak) in play here is the indefinite (continuous) execution of the division, but the resulting number of decimal places will always be finite.
Why can’t mathematical infinities of different sizes exist?
Answer: A numerical value cannot be infinite. The word “sizes” (logically speaking) when used to refer to the idea of actual infinities, is an incoherent use of language. Although the laws of mathematics don't limit the counting process, any number suggested can always be made into a larger finite value by adding 1... in fact, infinity isn't even a value!
Number sequences such as the natural numbers [+1, +2, +3, etc.] vs the real numbers [-0.1, +0.27, -4.0, i.e., literally any numbers], will obviously result in a different quantity (cardinality; size) of numbers being present if each sequence is put into a set/container of a given entry value limit, e.g., limit = 3, results in: [1, 2, 3] and hypothetically: [0.5, 1, 1.5, 2, 2.5, 3].
However, when no value limit is imposed on such sets, what would actually result (in reality) after adding an entry to the natural numbers set each time the hypothetical real numbers set is added to, is: [1, 2, 3, 4, 5, 6].
Related analysis: “One-to-one correspondence of Cantor’s diagonal argument”. The point being – the absence of 1 to 1 correspondence does not mean that there actually exists different size limitless sequences, obviously... so why are people being taught this in such a way that they spend their lives believing it to be true? – Moreover, “limitless” can only exist as potential, i.e., things cannot have an actual infinite existence.
Even if platonic numbers did exist, these sequences: [1, 2, 3, 4, 5, 6 ...] and [0.5, 1, 1.5, 2, 2.5, 3 ...] would be the same infinite size (i.e., undefined, as always), or indeed any other sequence, including the natural numbers... but remember, neither abstract platonic numbers, nor infinity, exists – also follow the links further down...
Notice that the previous two sequences each contain 6 entries (or 6 + infinity “...” for the “dot” believers), and if numbers are added to the lists indefinitely, they'll simply grow to always be at the same size. And because mathematical laws (so to speak) don't actualise mathematical objects into existence all by themselves, the best we can do to force a container to increase in size is to keep adding to such lists via fast computers, but the size of such lists will always be finite, even if the computers don’t run out of memory.
In contrast to the above, the following sequences must exist by the same logic: natural numbers [1, 2, 3, 4 ...] vs the odd numbers [1, 3] or [1, 3, 5, 7 ...]
But... set theory asserts the correspondence between these two sequences to be such that, they are indeed the same limitless size... in contrast to how the real numbers are treated... so ok then, simply 1 to 1 correspondence... but that's absolutely (logically) nothing to do with the true meaning of infinity – a misconception.
The video below presented by “Vsauce”, provides a nice overview (cool graphics) of the different stages of infinite existence as asserted by set theory.
See: 12:19 to 16:25 to get an appreciation of how axioms (“a statement or proposition on which an abstractly defined structure is based”) are simply assumed to be true.
Therefore, three key points are simply:
Sequences can be added to indefinitely.
Correspondence rules as used in set theory, do indeed identify useful relationships.
Axiomatic correspondence as used to assert the existence of different size infinite sets, disregards logic and common sense.
No set can actually contain an infinite quantity of numbers; using axioms, symbols, or mathematical notation as a way of declaring a set to be infinite in quantity, doesn’t materialise an actual infinity into existence (i.e., not infinity at all).
Distinguishing between the terms: “infinity” and “indefinitely” and not being misled into believing that notation such as “...” asserts (justifies) the existence of infinity, might help to avoid much of the confusion that exists with respect to the concept (idea) of infinity – a misconception.
Incidentally, platonic objects do not escape paradoxical numerical analysis: Potential infinity disproven (see: “Infinitesimal Paradoxes Identified”)
Interesting: Analysing how elements in sets correspond to each other.
Interesting: Identifying that they can indeed be treated as having different cardinalities (sizes) when 1 to 1 correspondence is applied.
Silly: Asserting that different size infinities actually exist.
Related video: “The Number-ness of Platonic Numbers” – “For Infinity: Professor Jim Franklin – Starts at 17:55”