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Some of these seem forced. For instance, does Chapernowne's number (number 7 on the list, 0.12345678910111213141516171819202122232425...) occur in nature, or wa

by drob518 10mo ago

Some of these seem forced. For instance, does Chapernowne's number (number 7 on the list, 0.12345678910111213141516171819202122232425...) occur in nature, or was it just manufactured in a mathematical laboratory somewhere?

zeeboo 10mo ago
It is indeed manufactured specifically to show the existence of "normal" numbers, which are, loosely, numbers where every finite sequence of digits is equally likely to appear. This property is both ubiquitous (almost every number is normal in a specific sense) and difficult to prove for numbers not specifically cooked up to be so.
- drob518 10mo ago
  Okay, fair. It just seemed to me to have pretty limited utility.
  - kaffekaka 10mo ago
    Hm who cares about utility in this case?
    - drob518 10mo ago
      Well, if we don’t care about utility I could define infinitely many transcendental numbers with no utility other than I just made them up. The number that is the concatenation of the digits of all prime numbers in sequence, for instance: 0.23571113171923… I christen this Dave’s Number. (It probably already has a name, but I’m stealing it.) Let’s add it to the list. Now we can define Dave’s Second Number as the first prime added to Dave’s Number: 2.235711131723… Dave’s Third Number is the second prime added to Dave’s Number: 3.235711131723… Since we’re cataloguing numbers with no utility, let’s add them all to the list.
      - kaffekaka 10mo ago
        The list was for famous numbers. Yours might get there, but not so fast.
        IAmBroom 10mo ago
        But it is definitely in the list of Aleph-1 Most Famous Transcendental Numbers. So there's that.
Strilanc 10mo ago
It's fame comes from the simplicity of its construction rather than its utility elsewhere in mathematics. For example, Graham's number is pretty famous but it's more of a historical artifact rather than a foundational building block. Other examples of non-foundational fame would be the famous integers 42, 69, and 420.
tantalor 10mo ago
Yes, it occurs in the nature of the mathematician's mind.
jerf 10mo ago
All the transcendental numbers are "manufactured in a mathematical laboratory somewhere". In fact we can tighten that to all irrational numbers are manufactured in a mathematical laboratory somewhere. You'll never come across a number in reality that you can prove is irrational. That's not necessarily because all numbers in reality "really are" rational. It is because you can't get the infinite precision necessary to have a number "in hand" that is irrational. Even if you had a quadrillion digits of precision on some number in [0, 1] in the real universe you'd still not be able to prove that it isn't simply that number over a quadrillion no matter how much it may seem to resemble some other interesting irrational/transcendental/normal/whatever number. A quadrillion digits of precision is still a flat 0% of what you'd need to have a provably irrational number "in hand".
- 5ver 10mo ago
  It appears quantum phenomena are accurately described using mathematics involving trig functions. As such we do encounters numbers in reality that involve transcendental numbers, right?
  - kevin_thibedeau 10mo ago
    They're accurately modeled. Just as Newtownian phenomena are accurately modeled, until they aren't. Reality is not necessarily reflective of any model.
  - jerf 10mo ago
    Consider the ideal gas law: pV=nRT Five continuous quantities related to each other, where by default when not specified we can safely assume real values, right? So we must have real values in reality, right? But we know that gas is not continuous. The "real" ideal gas law that relates those quantities really needs you to input every gas molecule, every velocity of every gas molecule, every detail of each gas molecule, and if you really want to get precise, everything down to every neutrino passing through the volume. Such a real formula would need to include terms for things like the self-gravitation of the gas affecting all those parameters. We use a simple real-valued formula because it is good enough to capture what we're interested in. None of the five quantities in that formula "actually" exist, in the sense of being a single number that fully captures the exact details of what is going on. It's a model, not reality. Similarly, all those things using trig and such are models, not reality. But while true, those in some sense miss something even more important, which I alluded to strongly but will spell out clearly here: What would it mean to have a provably irrational value in hand? In the real universe? Not metaphorically, but some sort of real value fully in your hand, such that you fully and completely know it is an irrational value? Some measure of some quantity that you have to that detail? It means that if you tell me the value is X, but I challenge you that where you say the Graham's Number-th digit of your number is a 7, I say it is actually a 4, you can prove me wrong. Not by math; by measurement, by observation of the value that you have "in hand". You can never gather that much information about any quantity in the real universe. You will always have finite information about it. Any such quantity will be indistinguishable from a rational number by any real test you could possibly run. You can never tell me with confidence that you have an irrational number in hand. Another way of looking at it: Consider the Taylor expansion of the sine function. To be the transcendental function it is in math, it must use all the terms of the series. Any finite number of terms is still a polynomial, no matter how large. Now, again, I tell you that by the Graham's Number term, the universe is no longer using those terms. How do you prove me wrong by measurement? All you can give me is that some value in hand sure does seem to bear a strong resemblance to this particular irrational value, pi or e perhaps, but that's all. You can't go out the infinite number of digits necessary to prove that you have exactly pi or e. Many candidates for the Theory of Everything don't even have the infinite granularity in the universe in them necessary to have that detailed an object in reality, containing some sort of "smallest thing" in them and minimum granularity. Even the ones that do still have the Planck size limit that they don't claim to be able to meaningfully see beyond with real measurements.
eichin 10mo ago
> mathematical laboratory Love the image of mathematicians laboring over flasks and test tubes, mixing things and extracting numbers... would have far more explosions than day-to-day mathematics usually does...
deleted 10mo ago
[deleted]

by drob518 10mo ago

zeeboo 10mo ago

It is indeed manufactured specifically to show the existence of "normal" numbers, which are, loosely, numbers where every finite sequence of digits is equally likely to appear. This property is both ubiquitous (almost every number is normal in a specific sense) and difficult to prove for numbers not specifically cooked up to be so.

drob518 10mo ago
Okay, fair. It just seemed to me to have pretty limited utility.
- kaffekaka 10mo ago
  Hm who cares about utility in this case?
  - drob518 10mo ago
    Well, if we don’t care about utility I could define infinitely many transcendental numbers with no utility other than I just made them up. The number that is the concatenation of the digits of all prime numbers in sequence, for instance: 0.23571113171923… I christen this Dave’s Number. (It probably already has a name, but I’m stealing it.) Let’s add it to the list. Now we can define Dave’s Second Number as the first prime added to Dave’s Number: 2.235711131723… Dave’s Third Number is the second prime added to Dave’s Number: 3.235711131723… Since we’re cataloguing numbers with no utility, let’s add them all to the list.
    - kaffekaka 10mo ago
      The list was for famous numbers. Yours might get there, but not so fast.
      - IAmBroom 10mo ago
        But it is definitely in the list of Aleph-1 Most Famous Transcendental Numbers. So there's that.

Strilanc 10mo ago

It's fame comes from the simplicity of its construction rather than its utility elsewhere in mathematics. For example, Graham's number is pretty famous but it's more of a historical artifact rather than a foundational building block. Other examples of non-foundational fame would be the famous integers 42, 69, and 420.

tantalor 10mo ago

Yes, it occurs in the nature of the mathematician's mind.

jerf 10mo ago

All the transcendental numbers are "manufactured in a mathematical laboratory somewhere". In fact we can tighten that to all irrational numbers are manufactured in a mathematical laboratory somewhere. You'll never come across a number in reality that you can prove is irrational. That's not necessarily because all numbers in reality "really are" rational. It is because you can't get the infinite precision necessary to have a number "in hand" that is irrational. Even if you had a quadrillion digits of precision on some number in [0, 1] in the real universe you'd still not be able to prove that it isn't simply that number over a quadrillion no matter how much it may seem to resemble some other interesting irrational/transcendental/normal/whatever number. A quadrillion digits of precision is still a flat 0% of what you'd need to have a provably irrational number "in hand".

5ver 10mo ago
It appears quantum phenomena are accurately described using mathematics involving trig functions. As such we do encounters numbers in reality that involve transcendental numbers, right?
- kevin_thibedeau 10mo ago
  They're accurately modeled. Just as Newtownian phenomena are accurately modeled, until they aren't. Reality is not necessarily reflective of any model.
- jerf 10mo ago
  Consider the ideal gas law: pV=nRT Five continuous quantities related to each other, where by default when not specified we can safely assume real values, right? So we must have real values in reality, right? But we know that gas is not continuous. The "real" ideal gas law that relates those quantities really needs you to input every gas molecule, every velocity of every gas molecule, every detail of each gas molecule, and if you really want to get precise, everything down to every neutrino passing through the volume. Such a real formula would need to include terms for things like the self-gravitation of the gas affecting all those parameters. We use a simple real-valued formula because it is good enough to capture what we're interested in. None of the five quantities in that formula "actually" exist, in the sense of being a single number that fully captures the exact details of what is going on. It's a model, not reality. Similarly, all those things using trig and such are models, not reality. But while true, those in some sense miss something even more important, which I alluded to strongly but will spell out clearly here: What would it mean to have a provably irrational value in hand? In the real universe? Not metaphorically, but some sort of real value fully in your hand, such that you fully and completely know it is an irrational value? Some measure of some quantity that you have to that detail? It means that if you tell me the value is X, but I challenge you that where you say the Graham's Number-th digit of your number is a 7, I say it is actually a 4, you can prove me wrong. Not by math; by measurement, by observation of the value that you have "in hand". You can never gather that much information about any quantity in the real universe. You will always have finite information about it. Any such quantity will be indistinguishable from a rational number by any real test you could possibly run. You can never tell me with confidence that you have an irrational number in hand. Another way of looking at it: Consider the Taylor expansion of the sine function. To be the transcendental function it is in math, it must use all the terms of the series. Any finite number of terms is still a polynomial, no matter how large. Now, again, I tell you that by the Graham's Number term, the universe is no longer using those terms. How do you prove me wrong by measurement? All you can give me is that some value in hand sure does seem to bear a strong resemblance to this particular irrational value, pi or e perhaps, but that's all. You can't go out the infinite number of digits necessary to prove that you have exactly pi or e. Many candidates for the Theory of Everything don't even have the infinite granularity in the universe in them necessary to have that detailed an object in reality, containing some sort of "smallest thing" in them and minimum granularity. Even the ones that do still have the Planck size limit that they don't claim to be able to meaningfully see beyond with real measurements.

eichin 10mo ago

> mathematical laboratory Love the image of mathematicians laboring over flasks and test tubes, mixing things and extracting numbers... would have far more explosions than day-to-day mathematics usually does...

deleted 10mo ago

[deleted]