πŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Dec 1

1/7
This air-quote #Mathematics unquote article https://www.quantamagazine.org/a-new-bridge-links-the-strange-math-of-infinity-to-computer-science-20251121/ keeps appearing in my feed, and I initially made some comments the first time, debunking it from a #Maths point-of-view, but given how it keeps popping up I think I need to do a more thorough #MathsMonday thread about it

Firstly, the author is a Physics journo, so you can take what he says about #math with a grain of salt (for some reason I see a lot of them doing this overreach, instead of checking with a Mathematician)…

Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Dec 1

2/7
"the strange infinite ones that other mathematicians ignore" - Mathematicians "ignore" them because there's no such thing

"Set theory deals with the infinite" - no it doesn't. Sets are finite, literally have an end. They have cardinality - the number of elements in the set. You can't have a cardinality of "infinity", nor is it a set if there's no closing bracket (which there can't be if it's infinite)…

Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Dec 1

3/7
"Georg Cantor, who proved in 1874 that there are different sizes of infinity" - no he didn't. I saw an article just recently about how other Mathematicians didn't take him seriously (and something about him thus taking his ideas to the Catholic Church instead)

In the linked article https://www.quantamagazine.org/how-can-some-infinities-be-bigger-than-others-20230419/ we find this gem, "physicists are sort of treating infinity in various sort of cavalier ways... and maybe not being as accountable for it as a mathematician would like them to be"...

How Can Some Infinities Be Bigger Than Others? | Quanta Magazine

All infinities go on forever, so how is it possible for some infinities to be larger than others? The mathematician Justin Moore discusses the mysteries of infinity with Steven Strogatz.

Quanta Magazine
Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Dec 1

4/7
"At the time, mathematicians were deeply uncomfortable with this menagerie of different infinities" - at the time other Mathematicians just ignored him, knowing that you can't have an infinite set

"While the set of real numbers between zero and 1 and the set of real numbers between zero and 10 are both infinite" - nope, they're both finite, by definition (end, are bounded), but this is a separate topic I'm going to cover more in-depth in a forthcoming MathsMonday…

Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Dec 1

5/7
"have the same cardinality" - no they don't, as per the subsequent part of the comment, "the first has a Lebesgue measure of 1 and the second a Lebesgue measure of 10"

"You’ll get an infinite number of connected nodes" - no you won't. The circumference of a circle is also finite. People (like this Physics major) keep throwing around the word "infinite" when they are discussing things which are very much finite, by definition. πŸ™„ Just saying it's "infinite" doesn't make it true...

Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Dec 1

6/7
"even if you have infinitely many sets to choose from" - which you don't. Again, the circumference of a circle is finite, by definition

"You’ve colored each node (which has zero length)... these sets are unmeasurable" - welcome to what happens when you pretend that a node can have a length of zero. πŸ™„ In which part of "edge/colouring" did you not understand that it has width?

"all just pieces of the circle’s circumference" - which has width, or you wouldn't be able to see the circle...

Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Dec 1

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"These infinite graphs are β€œuncountably” infinite" - he said, in a true truth πŸ˜‚ (spoiler alert: they weren't infinite anyway). BTW, this "countable" referred to, is people using the wrong word - what they ACTUALLY mean is discrete. e.g. the integers are discrete and infinite. You can't count something infinite, by definition

"This is a very interesting experience, trying to prove results in a field where I don’t understand even the basic definitions" - welcome to Cantor's world πŸ˜‚

Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Dec 8

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#Maths #Math
This #MathsMonday Why there isn't an #infinite amount of #numbers between 0 and 1 on the #numberline AKA The number-line isn't a TARDIS!

Firstly, one thing to point out is that people often conflate numbers with #numerals. This is important because #scalars also use numerals. Even though both numbers and scalars use numerals, they aren't the same thing!

In #Mathematics, especially #arithmetic, we use numbers to count things. They enumerate how many things there are...

Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Dec 8
2/7
Numbers, such as integers, are discrete. This means they are separate to each other, and there is nothing in-between. An example of discrete is clothing sizes, Small, Medium, and Large. There isn't a size between Small and Medium, and there isn't a size between Medium and Large. In the Integers, there is nothing between 1 and 2. When you bring Fractions into the mix, there is nothing between them either (for a given denominator - for hundredths, nothing between 1/100 and 2/100)…
Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Dec 8
3/7
Now imagine we put 12 eggs into an egg carton. Now say the eggs were hard-boiled, and we cut them into halves. Now we have 24 halves, but still 12 eggs total. Still as much as the carton can fit. Now, chop! chop! chop! chop! into millionths. Now the pieces are nearing infinitesimally small. We have 12 million millionths of egg pieces, and still 12 eggs total. There wasn't suddenly an infinite amount of pieces, nor were there more pieces, and still fits in the egg carton...
Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Dec 8

4/7
Hopefully at this point you can see no matter how much we chop up the eggs, we can never get an infinite amount of pieces - we're going to arrive at the molecular level before that happens, and what are we going to do then? Split the atoms?? We still can't get an infinite number of pieces to suddenly magically appear (though we may blow up our city with a nuclear blast πŸ˜‚)!

Now let's take that same example, but we'll cut into tenths each time...

Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Dec 8
5/7
So with our first lot of cuts we end up with 120 tenths of eggs. One tenth as a decimal is 0.1. Chop by tenths again, we now have 1200 hundredths of eggs, each piece 0.01egg in size. Keep going, thousandths 0.001, ten-thousandths 0.0001, etc. As we keep going smaller and smaller, we have a series which is approaching, BUT NEVER REACHES, 0, and we still have the same total of 12 eggs that we started with (as long as we didn't blow the place up along the way πŸ˜‚)!...
Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Dec 8
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And that remains true even as we continue on to an "infinite" number of decimal places (it's a hyperbola with an asymptote of 0) - as long as the number is non-zero, it takes up a non-zero amount of space, and in fact the total still takes up the EXACT amount of space that we started with - a carton of 12 eggs (now in infinitesimally small pieces)! You literally CANNOT fit an infinite amount of numbers into a finite part of the number-line. You can't make them small enough to become 0...
Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Dec 8

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And 0 is the ONLY thing that you can add an infinite amount of times, and the further issue there is that in our number-line interval - 0-12 (12 eggs) - 0 appears ONCE, at the start, so you STILL can't fit an infinite amount of numbers into that finite space. The number-line IS NOT a TARDIS! πŸ˜‚ A finite space can ONLY contain a finite amount of non-zero things. Non-zero numbers, even "infinitely" long ones, take up non-zero space.

To be continued (more on scalars next time)

Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Jan 5
1/5
This #MathsMonday I'm expanding on #scalars in #Mathematics. Previously we discussed that integers in #Maths are discrete, and used for counting things. In #Math we also have the Real Numbers, which is a bit of a misnomer since we don't usually use them for counting! It's more a case of people conflating them with numbers, because both use numerals. Rather, the Reals are used as multipliers, to provide a scale (though sometimes used as a fraction's decimal equivalent)…
Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Jan 5
2/5
I've talked about this in explaining Terms https://dotnet.social/@SmartmanApps/111084302312270707 where I used the example of someone 2 metres (2m) tall. They're not 2 1m halves stuck together, they're a single person, of a height in units of metres multiplied by 2 - the 2 is being used as a scalar, not an enumerator (number). With that in mind, consider pi. Pi isn't a number - we aren't counting some discrete things - we're using it as a scalar - we multiply the diameter of a circle by pi to find the circumference...
πŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“± (@[email protected])

Attached: 1 image 1/6 #MathsMonday I continue to see people who say that in #Mathematics ab=a*b, and thought of a good #Maths example to illustrate why ab is a single #Math Term (i.e. ab=(a*b))... Let's say I was 2 metres tall (just for the sake of using whole numbers in the example). We write that as 2m - in this case m is short for metres, but it also looks like an algebraic term, right? πŸ™‚ So let's say m is a pronumeral, and in this case m is equal to 1 metre. In other words in both cases, m is the units...

dotnet.social
Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Jan 5

3/5
Pi is relevant to this discussion, because it is non-terminating. i.e. it has an infinite number of decimal places. There are 2 points to be made here...

1. we don't put scalars on the number-line, we use it for, you know, numbers - it's right there in the name! πŸ˜‚

2. getting to the crux of the matter, there are people who say that because you can have an infinite amount of decimal places for each number, you can fit an infinite number of them into the interval 0-1 on the number-line...

Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Jan 5
4/5
No, you can't fit an infinite amount of them into a finite interval! As we already saw https://dotnet.social/@SmartmanApps/115682175610484093 you CANNOT fit an infinite amount of ANYTHING into a finite thing. You can fit an infinite amount of 0, but you cannot fit an infinite amount of non-zero numbers, so no matter how many decimal places it has, even "infinite", it still takes up a non-zero amount of space on the number-line, and you can thus only fit a finite amount of them into a finite interval...
πŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“± (@[email protected])

Attached: 1 image 1/7 #Maths #Math This #MathsMonday Why there isn't an #infinite amount of #numbers between 0 and 1 on the #numberline AKA The number-line isn't a TARDIS! Firstly, one thing to point out is that people often conflate numbers with #numerals. This is important because #scalars also use numerals. Even though both numbers and scalars use numerals, they aren't the same thing! In #Mathematics, especially #arithmetic, we use numbers to count things. They enumerate how many things there are...

dotnet.social
Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Jan 5

5/5
So next time you see someone say "you can have infinite decimal places, so we can fit an infinite amount of them into the interval 0-1", just point them to this thread. The number-line isn't a TARDIS!

P.S. As a third point, pretty much all scalars are only accurate to 2 or 3 decimal places anyway! Times (in sports events) are to hundredths or thousandths of a second, weights to thousandths of a kilogram (grams), etc., there's no measuring device accurate to infinite decimal places πŸ™„

Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±

1/9
Coming back this #MathsMonday to Cantor (who has regrettably been filling my #Mathematics timeline again), we have https://www.quantamagazine.org/how-can-infinity-come-in-many-sizes-20260223/ which lays out his (non-)proof in layman's terms, so let's look at the specifics of the #Maths...

"Aristotle rejected the existence of the infinite entirely; to him, infinity was simply a limit that could never be reached, not a true mathematical entity" - yep, and this is what is still taught about #Math limits and infinity today...

Mar 9 at 7:11amWeb
Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Mar 9

2/9
"dismissed his work as that of a madman" - we're about to see why! πŸ˜‚

"form the bedrock of modern mathematics" - no, the bedrock of Maths is still arithmetic, as taught at school

"we’re actually matching the natural numbers (1, 2, 3, and so on) to each object" - which doesn't need sets

"The set of natural numbers appears to be twice the size of the set of even numbers" - neither is a set (not finite), and "size" has no meaning if you are comparing things which aren't finite...

Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Mar 9

3/9
We need an end to say WHICH end is further away (bigger). They've said like a 2 second musical tone of twice the frequency of another is also twice as long - it isn't. The Natural numbers have twice the frequency of the evens, but both are infinite

"match each natural number to each even" - and yet, at any point, only be half as far away from 0 on the number line - if we stopped (at say 100), the Naturals set would be twice as big, with half being unmatched, a crucial overlooked fact... πŸ™„

Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Mar 9

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"Conclusion: This complete matching means that both sets are the same size" - conclusion: you haven't understood the previous points, thus arriving at a wrong conclusion...

β€œcountably” - as already discussed, "discrete" is the word you're looking for

"there are infinitely many rationals just between the numbers zero and 1" - as already discussed, no there isn't. You can only fit a finite number of things into a finite space...

Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Mar 9

5/9
"If you match each natural number... you’ll never get to the second row" - which you never do anyway, given the first row is, you know, infinite πŸ˜‚

"Conclusion: Once again, the sets are the same size" - once again, a wrong conclusion

β€œuncountably” - as discussed, "continuous" is the word you are looking for. This is like comparing eggs in a carton (discrete) to the possible settings of a volume knob (continuous)... πŸ™„

Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Mar 9

6/9
"real numbers, which includes all fractions as well as numbers like √2 and Ο€" - as discussed, the (irrational) Reals aren't numbers (aren't discrete, nor used for counting) - √2 and Ο€ are both scalars. You can't count 1, √2, 2 things - both these exist only in geometry (such as circles and Pythagoras), Physics (acceleration), etc.

"what happens when we compare them" - yes, let's compare eggs in a carton with volume knob settings... πŸ™„

Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Mar 9

7/9
"We’re going to prove that this will never work" - it's because the Reals are continuous, not discrete like numbers. Many Mathematicians knew that already for centuries before πŸ™„

"Examine your list of all real numbers... The new number you’ve generated can’t be in your original list" - if it wasn't in your list, then your list wasn't "all real numbers". Both things CAN'T be true - either it was in your list, or your list WASN'T "all real numbers" to begin with...

Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Mar 9

8/9
"Conclusion: The set of real numbers must be larger than the set of natural numbers" - conclusion: there's more possible settings on a volume knob than eggs in a carton. Spot the difference πŸ™„

"By doing this for every number in the first set, you’ll get a one-to-one match" - no, you won't. 0->0, 0.1->0.2, 0.2->0.4,... 0.1 isn't matched, 0.3 isn't matched, we in fact have a pattern in the second set of matched, unmatched, matched, unmatched - are we seeing yet why they said he's a madman? πŸ˜‚

Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Mar 9

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"Conclusion: The sets are the same size" - conclusion: you're a madman who managed to miss that half the numbers in the second set are unmatched

"In fact" - they said, in a completely fact-free post πŸ™„

"real number line" - there are no Reals on the number-line, only numbers - it's right there in the name! πŸ˜‚

"infinite sets can defy our intuition" - should've listened to your intuition

"called into question what mathematicians thought they knew" - no, they called his sanity into question