Not sure what you’re getting at here, particularly, from previous posts you’ve made. Let me restate:
Our present existence can be summed-up by the term ‘determinism’; ie, it’s a generally held belief that all of creation is governed by rules which, one day, can be explained mathematically by us humans. There is no randomness in nature… but hold on, mathematics seems to show that the opposite is the case. Just look at the Prime Numbers, or the square root of two, or pi, or loads of others. These numbers don’t resolve themselves. They can’t be explained. They are random.
The Universe is random, which might blow apart everything us feeble humans think about it.
(by the way, ‘thought’ is also random. Ha! another story)
@RobG . I remember when at tech college, an imaginary number. Of course with the dumbing down of education, the tech college is now a university. I digress.
The imaginary number was the square root of minus 1. I can’t remember exactly what it was for but it allowed a formula which I thinks was to do with electrical power calculations to work. Without it, well nada I guess. As you said, there is randomness in nature.
It had nothing to do with my previous posts, but with your question what the square root of two is (so I gave you the definition) and wanted to know what “unresolved” means for a number (“These numbers don’t resolve themselves”). I don’t know what you mean by this (nor what it means for eg the square root of two to be “random”).
Bonjour Pat (and Willem). I believe they are known as ‘pseudorandom numbers’.
Algorithms can only work with predictability - a computer programme will always follow the same path to completion. A good example of this is random number generation (used for games of chance, and more importantly for generating passwords and encrypting data). It is fundamentally impossible to produce truly random numbers on any deterministic device. Instead, computers produce what are known as ‘pseudorandom numbers’, a stream of numbers that appear as if they were generated randomly; because, as Von Neumann said: “Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin”
A few years back websites began to push their cookies policies at visitors, supposedly aimed at improving privacy. Now it’s next to impossible to follow any link without the need to click some button/s “agreeing” to some or other yawnsome tripe that no one ever reads. Apart from the momentary irritation this produces I’ve never thought much about this, but is there some deeper reason for having to “agree” before proceeding? Your mention @RobG of scanning a QR code as a prerequisite for [increasingly more things, take your pick] reinforces that impression. It’s gradual but steady conditioning, a drill. Unthinkingly acceding to minor inconveniences makes the major ones more likely, I think. And this has certainly been accelerated since CronyVirus came onto the scene.
The development of AI is interesting and potentially useful, I suppose, but the results so far seem eerily consistent. To interact with the bot, the telephone tree, the self-service checkout, you have to learn the rules the bot needs you to follow. Frequently I want to lift up my shopping bag and rearrange the contents, so that I can fit in a couple more items, but the self-service checkout is programmed to find this suspicious and to whine for assistance. So I curb my own innocent behaviour.
This is precisely the self-policing, in a carceral society, that Foucault warned about.
It’s only a mask. Just two more weeks. I get a test for you.
Willem, I haven’t been explaining myself very well here. I’ll have another go:
As mentioned, the digital computer is a deterministic device; ie, it can only follow a series of steps (put there by humans) which we can call ‘a programme’, or a language.
The language is incredibly simple Binary, 1s and 0s, even though computers may appear to do amazing things (like predicting climate change and pandemics - whoa, don’t start getting political!).
The digital computer works on very simple maths (but it’s actually quite brilliant the way it’s done).
But the thing is, maths is a very flawed language.
Many numbers are unresolvable. They can’t be worked out. The square root of 2 is a good example of this. There is no answer to it at our present level of understanding.
Bottom line is, the digital computer is a very clever device, but it uses a flawed language (maths) and thus anything a digital computer spits out should be taken with a degree of skepticism.
The one I always liked was: what’s a third of 100?
33.333333333333333333333etc; ‘recurring’, just going on forever, our maths lecturer Alaric Howland would say glibly; knowing that that’s not a real answer. The number is indeed irresolvable.
I’ll let Willem do the heavy lifting as he’s forgotten more about maths than I’ve ever learned.
The way I think about some of your points is that the number is a real thing, but how we choose to write it down depends on culture and circumstance.
The Romans, for example, didn’t have a straightforward number system to write fractions (afaik), but it’s quite clear that halves and quarters still existed. Just because they couldn’t write them down easily didn’t mean the numbers weren’t real…same can be said for the Egyptians etc.
The same is also still true for the particular way we choose to write down numbers now. We might come up with a different set of mathematical symbols in the future (or if we ever met our alien overlords) that has a simple representation of the square root of two, but whether we can write it down easily or not has nothing to say about the existence of a number that when multiplied by itself equals two.
It’s like saying that we can’t accurately capture the sound of a blackbirds in musical notation, so it doesn’t exist…
On the point about computers, as I’ve spent about half my life now building mathematical models with computers, I can’t say that I agree with you completely, but you have something of a point. All outputs from a computer model should be carefully checked against reality before you start to believe in them.
Just like the scientists do, very carefully, and endlessly, with climate change.
I think I understand what you’re saying. I expected you were talking about representing the square root of two (or other numbers) by their decimal or binary expansions, but it wasn’t clear.
Yes, as PP says, these numbers are fixed and have a precise meaning; it’s just that if you use sums of powers of 2 or 10 to represent them, they are necessarily not finite. It’s a “fault” of using this system, not with the number, which exists independently of any particular representation.
In the case of root 2, it happens because the number is irrational. In the case of 1/3, it’s because 3 doesn’t divide into 2 and 5. Note that in base 3, the number 1/3 would be represented by 0.1 (which perhaps you’d call “resolvable “).
But numbers can be represented differently; eg root 2 is the length of the diagonal of a square of side 1; or
Whatever number system you use you will encounter what mathematicians call ’ the lazy eight’. I’ll try to give an image of it here:
‘The lazy eight’ is the mathematical term/symbol for infinity (recurring numbers, etc), and it’s used in many complex equations (in order for the equations to reach any kind of rationality).
I find it wonderful that a system that can only function in a deterministic way has to acknowledge ‘infinity’ in order to work properly.
It’s enough to make you believe in God; or at least, to sit in a deckchair and get drunk.
Ah yes, infinity, though in my few decades in mathematics I’ve never heard of the ‘lazy eight’. You learn something new every day.
But there’s nothing lazy about infinity as the concept is used for limits. A lot of work (hundreds of years) went into perfecting this (subtle) definition. My “ad infinitum” contains a very strict definition* and is an amazing shorthand.
The only laziness I’ve noticed (among students usually) is an unwillingness to put the effort into understanding it. Instead one reads nonsense like “infinity-infinity =0”, as if infinity is a number.
Ah, now you’re talking. I can go along with that.
*For my particular limit, if a_n represents the nth term in the sequence, then, no matter how small we take e>0, there exists a whole number N such that for all n>N, a_n differs from root 2 by less than e.
Oh, and by the by, the ‘lazy eight’ now seems to have gone down the memory hole with search engines/big tech.
They don’t want you having any knowledge of this stuff. They want you to be using ‘smart phones’ and to be in the digital prison that you are handing to them.
You can demolish all this shite in two seconds flat, but most people don’t listen.
