Clinamenic

θερμοδυναμικός καπιταλιστής
I dunno.

Don't get too hung up on this I think. They were invented to get past an impasse but as soon as they were out of the box then they took on a life of their own. They are as computable as anything else now, probably best seen as one of a list of types of numbers that were called into existence as they were needed (see post above).
Yeah "computable" may not be the best word to describe the sort of numbers that i seems to constitute an exit from, but then again I wouldn't say "real" makes much sense either. I guess I just mean conceivable, at least for me. I cannot conceive of the square root of -1, but you're right that doesn't mean I cannot compute it via i as a sort of proxy.
 

IdleRich

IdleRich
Yeah "computable" may not be the best word to describe the sort of numbers that i seems to constitute an exit from, but then again I wouldn't say "real" makes much sense either. I guess I just mean conceivable, at least for me. I cannot conceive of the square root of -1, but you're right that doesn't mean I cannot compute it via i as a sort of proxy.
Have you seen complex numbers represented as points on a grid? Maybe that is a good way to represent them as they cannot lie anywhere on the basic number line that you meet at school. The idea of an imaginary number line orthogonal to the usual one is a neat one I think.
Can you conceive of irrational numbers though? I am not sure that I can very well... in a way I find them harder to think about than complex numbers.
 

Clinamenic

θερμοδυναμικός καπιταλιστής
So invention not discovery?
Does it ever feel like you are discovering something within? Socrates says something to Meno along these lines, but he described it as tapping into some internal wealth of potential knowledge. (edit: unrealized knowledge, that is)

Perhaps the extent to which [it feels like] you work to reach an understanding, rather than that understanding being felt as dawning upon you, is the extent to which it feels like invention rather than discovery.

I think it can be framed both ways, and we needn't choose. I tend to take this approach, just pragmatically switching between possible conceptions in situations wherein no single conception strikes me as most useful.

edit: bracketed text
 
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Clinamenic

θερμοδυναμικός καπιταλιστής
Have you seen complex numbers represented as points on a grid? Maybe that is a good way to represent them as they cannot lie anywhere on the basic number line that you meet at school. The idea of an imaginary number line orthogonal to the usual one is a neat one I think.
Can you conceive of irrational numbers though? I am not sure that I can very well... in a way I find them harder to think about than complex numbers.
Is pi an irrational number? Are those the ones what cannot be expressed as fractions? Pi is still mysterious to me, but I do understand its significance to some extent, its relation to a circle with a radius of 1 unit, etc.

The more elaborate characteristics and applications of pi are still beyond my comprehension though.

Also have trouble with the basic trig functions. I can visualize the unit circle with a tangent rotating around it, the intersections with the axes accelerating off to infinity before looping back from the opposing infinity, but I'm not sure how to understand the basic functions in these terms, or if such a visualization would even be useful in doing so.
 

Clinamenic

θερμοδυναμικός καπιταλιστής
So invention not discovery?
Could well be that we discovered something fundamental to our cognitive modality, which could have implications for [our understanding of] a more primordial nature than our own.

edit: bracketed text
 

Clinamenic

θερμοδυναμικός καπιταλιστής
In my mind, the concept of invention implies an origination that is difficult to qualify. As opposed to an ongoing reappropriation and recombination of components, wherein origination is only a concept.
 

Clinamenic

θερμοδυναμικός καπιταλιστής
I think it can be framed both ways, and we needn't choose. I tend to take this approach, just pragmatically switching between possible conceptions in situations wherein no single conception strikes me as most useful.
For the record, this could be considered a major, practical takeaway from Deleuze-Guattari, namely all the talk of schizo, as opposed to needing to boil things down to the one. But here, again, I don't feel the need to choose. The need to choose one would be a higher order repetition of the same instinct that schizo seems to oppose.
 

IdleRich

IdleRich
Is pi an irrational number? Are those the ones what cannot be expressed as fractions? Pi is still mysterious to me, but I do understand its significance to some extent, its relation to a circle with a radius of 1 unit, etc.

The more elaborate characteristics and applications of pi are still beyond my comprehension though.

Also have trouble with the basic trig functions. I can visualize the unit circle with a tangent rotating around it, the intersections with the axes accelerating off to infinity before looping back from the opposing infinity, but I'm not sure how to understand the basic functions in these terms, or if such a visualization would even be useful in doing so.
Basically at one point they thought they had all the numbers that existed - whole numbers, negative numbers, fractions (ie ratios or rational numbers) and that was it. If something wasn't a whole number then they believed it could be written as a ratio.
But then you got Pythagoras, his famous theorem about the square on the hypotenuse, and someone asked "if you have a right angled triangle with both the short sides having length one, then what is the length of the hypotenuse?" and the quick answer it is 1 plus 1 and then square rooted ie root 2 but what IS that? What fraction?
And they buggered about for ages trying to find the relevant fraction until some clever clogs was able to prove that no fraction would possibly work ie the square root of two couldn't be written as a fraction so it was a number which was outside of all those they believed to exist. Allegedly the guy who discovered this was murdered for his heresy but I think that may be just a myth.
Anyway, as they went on they discovered more irrational numbers - yes pi etc - and then later Cantor showed that there couldn't be a one-to-one mapping between rational numbers and irrational ones, in other words there are in sense MORE irrational numbers than rational numbers, both are infinite in number but one infinity is bigger than the other.
So most numbers are irrational, there is an irrational number between any two rational numbers you name, in fact there is an irrational number between any two irrational numbers.... infinitely many even. But WHAT exactly are they? I Find them so hard to imagine cos each one can only be expressed as an infinitely long decimal expansion (or a name) but for me it makes it hard to pin them down.
 

woops

is not like other people
- and then later Cantor showed that there couldn't be a one-to-one mapping between rational numbers and irrational ones, in other words there are in sense MORE irrational numbers than rational numbers, both are infinite in number but one infinity is bigger than the other
Was this his diagonal argument? I was raised by a mathematician, but I was hopeless at maths, much to his disappointment
 

IdleRich

IdleRich
Was this his diagonal argument? I was raised by a mathematician, but I was hopeless at maths, much to his disappointment
Oh yeah the one where you draw the numbers as kinda like a grid? You can prove it that way for sure, not sure if that was how Cantor did it, but if you say it was then I'm sure you are right. Edit you're right he did do that.
 
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Clinamenic

θερμοδυναμικός καπιταλιστής
Basically at one point they thought they had all the numbers that existed - whole numbers, negative numbers, fractions (ie ratios or rational numbers) and that was it. If something wasn't a whole number then they believed it could be written as a ratio.
But then you got Pythagoras, his famous theorem about the square on the hypotenuse, and someone asked "if you have a right angled triangle with both the short sides having length one, then what is the length of the hypotenuse?" and the quick answer it is 1 plus 1 and then square rooted ie root 2 but what IS that? What fraction?
And they buggered about for ages trying to find the relevant fraction until some clever clogs was able to prove that no fraction would possibly work ie the square root of two couldn't be written as a fraction so it was a number which was outside of all those they believed to exist. Allegedly the guy who discovered this was murdered for his heresy but I think that may be just a myth.
Anyway, as they went on they discovered more irrational numbers - yes pi etc - and then later Cantor showed that there couldn't be a one-to-one mapping between rational numbers and irrational ones, in other words there are in sense MORE irrational numbers than rational numbers, both are infinite in number but one infinity is bigger than the other.
So most numbers are irrational, there is an irrational number between any two rational numbers you name, in fact there is an irrational number between any two irrational numbers.... infinitely many even. But WHAT exactly are they? I Find them so hard to imagine cos each one can only be expressed as an infinitely long decimal expansion (or a name) but for me it makes it hard to pin them down.
Amazing. And to me this reflects certain limitations, which isn;t to imply a ceiling, to our mathematical mode of understanding.

Don't know the maths of Godel, but I have a basic grasp of the philosophy, that a system of knowledge cannot be consistent and complete, i.e. that any logically consistent rule has its blindspots, things it cannot account for. Such rules as the various categories of numbers you mention.

If we are talking about taxonomies of numbers, a nested taxonomy, whatever, then it would follow that every new category can only account for some subset of the numbers hitherto unnacounted for, no?

And clever definitions of new categories can delay the discernment of exceptions to those categories, but arguably not prevent them. Unless, perhaps, the rule is somehow inconsistent, in the interest of being complete, but that seems to run against the established grain of science as being a maximally consistent framework of knowledge.

You could classify all unclassified numbers under a genus of sui generis, in a paradoxical manner, but that seems useless.
 

IdleRich

IdleRich
Amazing. And to me this reflects certain limitations, which isn;t to imply a ceiling, to our mathematical mode of understanding.

Don't know the maths of Godel, but I have a basic grasp of the philosophy, that a system of knowledge cannot be consistent and complete, i.e. that any logically consistent rule has its blindspots, things it cannot account for. Such rules as the various categories of numbers you mention.

Well Cantor ideas I guess led to Godel and Incompleteness which is what you're talking about here.


If we are talking about taxonomies of numbers, a nested taxonomy, whatever, then it would follow that every new category can only account for some subset of the numbers hitherto unnacounted for, no?

I suppose. But are the unaccounted for numbers already out there to be discovered or do we just invent them? And does that change that?

And clever definitions of new categories can delay the discernment of exceptions to those categories, but arguably not prevent them. Unless, perhaps, the rule is somehow inconsistent, in the interest of being complete, but that seems to run against the established grain of science as being a maximally consistent framework of knowledge.
I don't know. Hopefully the rules won't be inconsistent. But you never know of course... to er is human.


You could classify all unclassified numbers under a genus of sui generis, in a paradoxical manner, but that seems useless.
This sounds like the start of the trick Russel used in his paradox (well sort of) and also the one that Godel and Turing used for their famous results. Creating these paradoxical sets turned out to be very handy.
 

Clinamenic

θερμοδυναμικός καπιταλιστής
I suppose. But are the unaccounted for numbers already out there to be discovered or do we just invent them? And does that change that?
If we consider the existing theories and classifications as a basis for further elaboration, then I suppose what we go on to understand can differ based on what basis we start from.

If we started from a different set of categories, we would have been predisposed to pursue different lines of inquiry, seeing as those categories would likely entail different exceptions and have different pain points.

So I would say that the unaccounted-for numbers that await us, as the known unknown, depend on our basis, which would be the known known. They are out there in that they are unrealized, but their realization could just as well be called invention as it could discovery.
 

Clinamenic

θερμοδυναμικός καπιταλιστής
I think the whole field of unrealized ideas is internal, and our navigation of it is partial to our basis of realized ideas, those that inform how we persist into the unrealized.
 
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