How did we get real numbers from rational number — T Clark

They were first discovered by the Pythagoreans. They were horrified by them though. They aggressively suppressed the knowledge of irrational numbers per legend. If it was an invention, it was not a welcome one. How do we explain that?

How did we get zero? — T Clark

This is a fascinating story involving the transcription of Babylonian abacus results.

This thread puts on display the way people try to escape from wonder. They assume a conclusion when they don't actually know any facts that support it. Psychic protection strategy?

Pi = the ratio of a circle's circumference to its diameter.

My mind tells me one of the main revelations of pi is the picture of the straight line of the diameter surrounded by the encircling circumference. This juxtaposition shows concisely that the rectilinearity (straight-lining) of science is only partially commensurable with the curvilinearity (curving) of nature.

The straight lines infinitesimal of the analysis of calculus can only approximate nature's reality.

Science is nature-adjacent rather than natural.

As technology diminishes and displaces nature, humanity rejiggers itself out of mysterious existence into self-reflection. The trick of AI and SAI is baking in a component of mystery and a component of error. Mystery and error support otherness, a component essential to forestalling the cognitive suffocation of an enclosing self-reflection.

Intentional mystery and error preserve the irrationality pictured by pi.

We must pull on and push against the idea our natural world is full mystery and error because some prior race of sentients understood the essential importance of forestalling cognitive suffocation. Having original sin in the mix is better than the damnation of perfection.

Against utopia! — ucarr

I'm unsure why this post hasn't gotten any replies, because this gets at the heart of the matter for why pi continues indefinitely.

A perfect circle simply doesn't exist. It can't be made by man, and not by machine. We can get close, but no matter how close we get, it will never be perfect, much like how a digital rendition of an analog signal can also never be perfect.

If we 'zoom in' one pixel (or one decimal) further, the imperfection shows.

What do we do with numbers like pi that go on forever? — frank

One third of 1 is 0.33333...........continuing to infinity.

If we altered our numbering system, such that we replaced 1 by 3, then one third of 3 is 1. This avoids any problem of infinity.

This suggests that the problem of infinity is an artificial problem of our numbering system.

Similarly with pi.

One third of 1 is 0.33333...........continuing to infinity.

If we altered our numbering system, such that we replaced 1 by 3, then one third of 3 is 1. This avoids any problem of infinity. — RussellA

I don't understand how we could replace 1 by 3. That doesn't make any sense. But with the new numbering system, 1/3 would be 1/5.

I don't understand how we could replace 1 by 3. — frank

If there is one object in the world, dividing it into three parts does not involve infinities. In our numbering system, dividing 1 by 3 does involve infinities.

This suggests that infinity is an artificial problem of our numbering system. Perhaps a different numbering system would avoid the problem of infinity altogether.

Changing notation does not remove the fact that π is an irrational number.

This is a fascinating story involving the transcription of Babylonian abacus results. — frank

I was fascinated by this, but I couldn't find anything specifically on it, although there are many versions available. On the other hand, this version does refer to accountancy, which does seem to me a practical application that is bound to trip over both 0 and negative numbers. (Both are needed to represent the critical difference between debit and credit and neither.)
Scientific American - Zero

This suggests that infinity is an artificial problem of our numbering system. Perhaps a different numbering system would avoid the problem of infinity altogether. — RussellA

I can see your point. but the ancient Greeks did not need the decimal system to prove that the square root of 2 or pi is irrational.
But, more fundamentally, if you define the numbers by reference to the operation "n+1", you already have infinity. Similarly "divide by 2" will also produce an infinite series, no matter what number system you have.

But I don't think that "invent" is the appropriate description. The story of the irrationals shows that when we set up the rules of a language-game (and that description of numbers is also an idealization), we may find that there are situations (applications of the rules) that surprise us. Hence it is more appropriate to say that we discover these — Ludwig V

If the rules of a language game make rational numbers intelligible, then isnt it a new set of rules that make irrationals intelligible? In other words, don’t we have to invent irrationals as well as rationals?

If the rules of a language game make rational numbers intelligible, then isnt it a new set of rules that make irrationals intelligible? — Joshs

It's a fiction that meaning arises from rule-following. There's no fact of the matter regarding what rules you've followed up til now.

This is a fascinating story involving the transcription of Babylonian abacus results.
— frank
I was fascinated by this, but I couldn't find anything specifically on it, — Ludwig V

Check out Zero: The Biography of a Dangerous Idea by Charles Siefe. It's pretty good.

If the rules of a language game make rational numbers intelligible, then isnt it a new set of rules that make irrationals intelligible?
— Joshs

It's a fiction that meaning arises from rule-following. There's no fact of the matter regarding what rules you've followed up til now. — frank

If we’re talking about Wittgenstein on rule-following here, then there is no intelligible meaning without rules, criteria, forms of life. But at the same time, in applying those concepts, criteria and rules, we don’t simply refer to them as a picture determining in advance how to go on. The rules underdetermine what to do in each new situation. There is an element of invention in following rules.

If we’re talking about Wittgenstein on rule-following here, then there is no intelligible meaning without rules, criteria, forms of life. — Joshs

The Private Language argument indicates that there's no way for you to know what rules you've been following up till now. Check out Wittgenstein on Rules and Private Language by Saul Kripke.

Or better, there's no fact of the matter about what rules you've been following.

If the rules of a language game make rational numbers intelligible, then isnt it a new set of rules that make irrationals intelligible? In other words, don’t we have to invent irrationals as well as rationals? — Joshs

The Pythagoreans denied their existence for a long time

It's a fiction that meaning arises from rule-following. There's no fact of the matter regarding what rules you've followed up til now. — frank

The Private Language argument indicates that there's no way for you to know what rules you've been following up till now. Check out Wittgenstein on Rules and Private Language by Saul Kripke. — frank

If the rules of a language game make rational numbers intelligible, then isnt it a new set of rules that make irrationals intelligible? In other words, don’t we have to invent irrationals as well as rationals? — Joshs

The Pythagoreans denied their existence for a long time after they realized the problem. No doubt they were working on arguments to establish that. They failed. It seems odd to describe that process as "inventing the irrationals". I don't know enough history to even comment on whether the rationals were invented or discovered. The number <omega> for the limit to an infinite series does look more like an invention to me. I don't know whether Cantor would agree with me.

It's a fiction that meaning arises from rule-following. There's no fact of the matter regarding what rules you've followed up til now. — frank

That's true, in a sense. But not the whole story.

If we’re talking about Wittgenstein on rule-following here, then there is no intelligible meaning without rules, criteria, forms of life. But at the same time, in applying those concepts, criteria and rules, we don’t simply refer to them as a picture determining in advance how to go on. The rules underdetermine what to do in each new situation. There is an element of invention in following rules. — Joshs

You state the problem nicely, but don't mention Wittgenstein's solution.

The Private Language argument indicates that there's no way for you to know what rules you've been following up till now. Check out Wittgenstein on Rules and Private Language by Saul Kripke. — frank

The PLA (insofar as it is an argument) establishes, IMO, that there is no way for you to know what rules you have been following up to now, if they are private rules. "Private" means that your say-so determines what is correct and what is not. So "correct" and "incorrect" have no application - no meaning.

What gives meaning to rules is human agreement in the context of human life. Think of how the fact that we agree on how to use words is enough to make them words. (This fact is, perhaps, not a fact of the matter, but it is a fact nonetheless.) What often gets left out of this is that we sometimes find that we don't agree on how to apply our rules; so we have to make a decision about how to go on.

What gives meaning to rules is human agreement in the context of human life. Think of how the fact that we agree on how to use words is enough to make them words. (This fact is, perhaps, not a fact of the matter, but it is a fact nonetheless.) What often gets left out of this is that we sometimes find that we don't agree on how to apply our rules; so we have to make a decision about how to go on. — Ludwig V

There's just nothing you can point to and say, "See, this is the rule I've been following for the use of this phrase."

The Private Language argument indicates that there's no way for you to know what rules you've been following up till now. Check out Wittgenstein on Rules and Private Language by Saul Kripke. — frank

There's just nothing you can point to and say, "See, this is the rule I've been following for the use of this phrase." — frank

Funny that this came up here just after I had used it in another thread.

Kripke misunderstood Wittgenstein's answer, found in PI §201

What this shews is that there is a way of grasping a rule which is not an interpretation, but which is exhibited in what we call "obeying the rule" and "going against it" in actual cases.

It's what we do that is of import. If Kripke were correct, you would not know how to count, yet you do know what it is to count, and by twos and threes as well as by ones. You understand what it is to carry on in the same way, and while you cannot say what this is, you cna show it by counting. This is the import behind the now cliched appeal: "Don't look to meaning, look to use".

If we’re talking about Wittgenstein on rule-following here, then there is no intelligible meaning without rules, criteria, forms of life. — Joshs

Don't look for an abstract thing called "the meaning". Look instead at what one is doing as a participant in the various activities that make up our daily lives. Then at least you will have a better idea of what Wittgenstein said.

You state the problem nicely, but don't mention Wittgenstein's solution.
The Private Language argument indicates that there's no way for you to know what rules you've been following up till now. Check out Wittgenstein on Rules and Private Language by Saul Kripke.
— frank
The PLA (insofar as it is an argument) establishes, IMO, that there is no way for you to know what rules you have been following up to now, if they are private rules. "Private" means that your say-so determines what is correct and what is not. So "correct" and "incorrect" have no application - no meaning. — Ludwig V

What gives meaning to rules is human agreement in the context of human life. Think of how the fact that we agree on how to use words is enough to make them words. (This fact is, perhaps, not a fact of the matter, but it is a fact nonetheless.) What often gets left out of this is that we sometimes find that we don't agree on how to apply our rules; so we have to make a decision about how to go on. — Ludwig V

It’s not human agreement , as though each individual voices their opinion and then the group arrives at a consensus. Socially normative meanings function prior to and already within individual experiences of rules and criteria of action. At the same time that such social norms allow us to make sense of our own perspective within them, we can differ among one another within shared language games as to how to proceed. And whether or not we agree on how to apply our rules, those rules never are enough to tell us how to go on. It is only within the actual context of the situation that we ‘intuit’ the specific sense and use of a rule. This intuitive knowing is the solution, not waiting for a consensus from a group.

Seems we pretty much agree, except that I don't think calling this an "intuition" is at all helpful, since it hints at private mental phenomena. It's not about intuition, it's about action - following a rule is something we do, not a "special sense".

But then I reject such a phenomenological approach.

It’s not human agreement , as though each individual voices their opinion and then the group arrives at a consensus. Socially normative meanings function prior to and already within individual experiences of rules and criteria of action. At the same time that such social norms allow us to make sense of our own perspective within them, we can differ among one another within shared language games as to how to proceed. And whether or not we agree on how to apply our rules, those rules never are enough to tell us how to go on. It is only within the actual context of the situation that we ‘intuit’ the specific sense and use of a rule. This intuitive knowing is the solution, not waiting for a consensus from a group. — Joshs

I agree with every word of that, except the word "intuit". But it's just a fancy name for the fact that we agree and usually, but not always, can resolve our disagreements on the basis of reasons, which, again, are reasons only because we are persuaded by them.
I would comment, though, that social norms, in this context, are not norms because they tell us what to do - that would make them just more rules; they are norms in the sense that we do in fact follow them for the most part. When we don't follow them, they cease to be norms.

There's just nothing you can point to and say, "See, this is the rule I've been following for the use of this phrase." — frank

If one could, it would just be another rule, and so not explain anything.
Yet, there are things we can point to and say "See, this is the rule I've been following". But that's because we have learnt the human practice of following rules, not because the rule tells us anything - apart from the words we put into its mouth.

Exactly, especially about Kripke.
In the end, it comes down to "This is what I do".

If Kripke were correct, you would not know how to count, — Banno

This shows a misunderstanding of Kripke's point. There's no denying that we do things with words, and that we do on occasion follow rules. There's just no fact regarding what rules you've been following up till now. If you think you know the rules you've been following, you need to take a closer look at the PLA.

Yet, there are things we can point to and say "See, this is the rule I've been following". — Ludwig V

I doubt it

This shows a misunderstanding of Kripke's point. — frank

I doubt it.

There's just no fact regarding what rules you've been following up till now. — frank

Sure, if what you mean is that the rule cannot be stated. But that is irrelevant, since the rule can be enacted.

Perhaps you need to take a closer look at the PLA.

Added: I'll fill that in a bit, rather than leave the implied but unintended offence. Kripke has his own idiosyncratic version of the private language argument. it is not generally accepted as what Wittgenstein argued for.

Sure, if what you mean is that the rule cannot be stated. But that is irrelevant, since the rule can be enacted. — Banno

There's no fact regarding which rules you've been enacting.

And yet we enact rules.

Where does "fact" fit here? What is a "fact"? And how does being or not being a "fact" fit in to enacting a rule?

If a fact is something we discover, find out abut, then it would be odd to think of what we might choose to do as being a fact... odd, for example, for you to say that you discovered that you had responded to my post. You didn't discover that response, it's just what you did. Sure, it's a fact you responded, but that's after the act. See the difference in direction of fit here? Following a rule is changing how things are to fit how you want them to be. Setting out a fact is changing what you say so that it matches how things are.

And yet we enact rules. — Banno

There's no fact regarding which rules. It's a mind bender for sure. It took me a good while to digest the implications.

There's no fact regarding which rules — frank

if a fact is something we discover. Not if a fact can be something we do. You know how to do plus, as opposed to quus. If you want, you might say that it is a fact that you do 2 plus 2 and not 2 quus 2.

And if you don't know which you are doing, then there's perhaps not much more to be said here, since I, and others, do understand what it is to follow plus and quus and to choose which to enact.

I, and others, do understand what it is to follow plus and quus and to choose which to enact. — Banno

Can confirm.

:smile:

Maybe even think of it this way: you know how to do plus or quus in the way you know how to ride a bike, not in the way you know that Sydney is in Australia.

That you cannot state what you do to ride a bike does not imply that you cannot ride a bike.

And riding a bike is not an "intuition".