I can't read the code present in Principa Mathematica, but I can vaguely grasp what's said about it on wikipedia. According to wiki, Wittgenstein had a few critiques of the book including the following:

Wikipedia wrote:It purports to reveal the fundamental basis for arithmetic. However, it is our everyday arithmetical practices such as counting which are fundamental; for if a persistent discrepancy arose between counting and Principia, this would be treated as evidence of an error in Principia (e.g. that Principia did not characterize numbers or addition correctly), not as evidence of an error in everyday counting.

Does anyone know if Wittgenstein (in his Lectures on the Foundations of Mathematics, Cambridge 1939) actually backs up that claim? Or does he just appeal to the popularity of the view that counting is "correct"? It seems to me that if it wasn't the former, than his criticism would not be very logically robust, as there could well be (in hypothesis) an error in everyday counting unless he was able to rigorously show otherwise.

If he DID rigorously show why any discrepency between the book and counting would be the result of an error in the book, could anyone explain (in English) how he went about showing that?

Secondly, he critiqued it with the following:

Wikipedia wrote:The calculating methods in Principia can only be used in practice with very small numbers. To calculate using large numbers (e.g. billions), the formulae would become too long, and some short-cut method would have to be used, which would no doubt rely on everyday techniques such as counting (or else on non-fundamental - and hence questionable - methods such as induction). So again Principia depends on everyday techniques, not vice versa.

This argument of "it's too much work" doesn't seem like a very valid critique at all.

Sityl wrote:Does anyone know if Wittgenstein (in his Lectures on the Foundations of Mathematics, Cambridge 1939) actually backs up that claim? Or does he just appeal to the popularity of the view that counting is "correct"? It seems to me that if it wasn't the former, than his criticism would not be very logically robust, as there could well be (in hypothesis) an error in everyday counting unless he was able to rigorously show otherwise.

If he DID rigorously show why any discrepency between the book and counting would be the result of an error in the book, could anyone explain (in English) how he went about showing that?

It wouldn't have surprised anyone if Principia had contained a mistake. Afterall, Principia was a response to a serious mistake Russell had identified in Frege's work.

Everyone knows how to count. Not everyone grasps the conceptual apparatus needed to count in Principia. So if there is a discrepancy, we'll blame the conceptual apparatus. Remember that texts like the Principia are, in part, about recovering what we already know. If we can't rightly recover the stuff we take to be obvious, such as methods of counting, then we've messed up. A marksman cannot claim that her shot was perfect, but the target was in the wrong place.

Secondly, he critiqued it with the following:

Wikipedia wrote:The calculating methods in Principia can only be used in practice with very small numbers. To calculate using large numbers (e.g. billions), the formulae would become too long, and some short-cut method would have to be used, which would no doubt rely on everyday techniques such as counting (or else on non-fundamental - and hence questionable - methods such as induction). So again Principia depends on everyday techniques, not vice versa.

This argument of "it's too much work" doesn't seem like a very valid critique at all.

I don't think that's his point. In wimpy systems, you spend a lot of your time reasoning outside the system. Sometimes we call this metalevel reasoning. Suppose you've written the same proof over and over, and you realise there's something general going on. But you might find that you can't express it in the system itself. Instead, you have to say things like "by induction on proofs", a claim made outside the system, invoking a principle that the system was intended to capture.

It's at this point we might feel a tension. How can the system be said to capture our everyday reasoning when much of the reasoning with the system is happening outside its scope?

Sityl wrote:there could well be (in hypothesis) an error in everyday counting unless he was able to rigorously show otherwise

Meaning what? What would an error in everyday counting look like?

An error is a violation of a norm, and Wittgenstein's point is that norms regarding counting are instituted by our practice of counting, not by some 20th century book most people never heard about (hence if the two differ, it is clear that the book hasn't captured those norms very well).

Thanks for the replies. An error would have to be something like:

Mathematica finds that there's an error in our counting which goldman sachs has been able to exploit to suck trillions of dollars out of the economy using simple, unequal exchanges that exploit this error.

Ok, I was kidding, but I guess, "it's just common sense" generally bothers me since common sense is so often wrong.

Does anyone know of an english translation to principa mathematica? (ie. "Maths" to "English" translation)

Sityl wrote:Thanks for the replies. An error would have to be something like:

Mathematica finds that there's an error in our counting which goldman sachs has been able to exploit to suck trillions of dollars out of the economy using simple, unequal exchanges that exploit this error.

Ok, I was kidding, but I guess, "it's just common sense" generally bothers me since common sense is so often wrong.

I think you've misunderstood Wittgenstein's point if you think the answer is "it's just common sense". It's as if you asked how we know there's no error in the rules of chess. No-one would answer "it's just common sense", the proper answer is a blank stare, or, if you're more philosophically inclined, to demand an explanation of what it means to say that there is an error in the rules of chess. Surely it would be ridiculous if I claimed to have discovered that we are mistaken about the rules of chess and that in fact bishops do not move diagonally, not because it goes against common sense, but because it is entirely unclear what such a claim might possibly mean.

Preno wrote:

Sityl wrote:Thanks for the replies. An error would have to be something like:

Mathematica finds that there's an error in our counting which goldman sachs has been able to exploit to suck trillions of dollars out of the economy using simple, unequal exchanges that exploit this error.

Ok, I was kidding, but I guess, "it's just common sense" generally bothers me since common sense is so often wrong.

I think you've misunderstood Wittgenstein's point if you think the answer is "it's just common sense". It's as if you asked how we know there's no error in the rules of chess. No-one would answer "it's just common sense", the proper answer is a blank stare, or, if you're more philosophically inclined, to demand an explanation of what it means to say that there is an error in the rules of chess. Surely it would be ridiculous if I claimed to have discovered that we are mistaken about the rules of chess and that in fact bishops do not move diagonally, not because it goes against common sense, but because it is entirely unclear what such a claim might possibly mean.

Thanks, that gives me a lot to think about. I guess I was thinking of counting as an entity in reality, rather than a construct that we use.

I don't suppose you have that maths to english translation?

I think Wittgenstein's point would be that one orange and one orange make two oranges is experientially obvious and sufficient to explain counting without having to try and prove it, he meant that since the number of oranges was a concrete and indisputable thing that any discussion on the topic was either trivial or if too involved susceptible to error.

What Wittgenstein perhaps didn't appreciate or care about (unless he was just saying it to be controversial, he was a very clever man) is that to explore virtual realms where we can't access oranges or apples we need to have a firm footing logically about what we mean when we talk about numbers and the operations that we use to manipulate them and what those operations can tell us about those realities.

The Principia could also tell us something about what happens to one orange when you perform certain operations on it. What happens if you start with one orange and you want to infer an orchard from it? How can you rigorously add that orange to itself in such a way that you know that when you then take either of the oranges away you are left with an orange that is the same as the one that you started with?

I haven't read it, but I was under the impression that the point of Principia was not to confirm the correctness of basic arithmetic on a firm footing but to put mathematics itself on a firm footing. This doesn't matter when you are dealing with basic arithmetic because intuition is the best guide there, but it becomes critical when you are dealing with concepts involving meta-arguments with several levels of abstraction, and intuition is nowhere to be found. The problem with only understanding arithmetic at an intuitive level is that you then have no basis for extending it to more abstract concepts. For instance, how will you know what constitutes a valid mathematical proof (I sometimes find myself wondering that myself when reading a particularly abstract or difficult proof)?

So yes, if there were a discrepancy between our understanding of basic counting and what Principia said, Principia would need to be corrected, but that's because Principia is using our understanding of counting as a guide to how to construct the foundations of mathematics and logic. It doesn't mean Principia is useless, because the point of Principia isn't to prove that counting is correct. It's to establish a basis for soundly extending our reasoning beyond counting.

But, I repeat, I haven't read it. That's just the impression I've formed from what other people have said about it.

I haven't read Wittgenstein's critique and only portions of the Principia Mathematica. However, when people talk about counting, I wonder things like how many eggs is a double-yolked egg? This is similar to other ideas like how many photons are there, or how many electrons there are within a 95% electron density surface. Also, I usually have a half an onion around the house which is more onion than some whole onions. I'm not sure that counting is as basic an idea as it's made out to be.

Wittgenstein's Philosophy of Mathematics: http://plato.stanford.edu/entries/wittg ... athematics

Sityl wrote:

Does anyone know of an english translation to principa mathematica? (ie. "Maths" to "English" translation)

http://www.guardian.co.uk/books/2009/ap ... nd-russell Enjoy

I haven't read it, but I was under the impression that the point of Principia was not to confirm the correctness of basic arithmetic on a firm footing but to put mathematics itself on a firm footing. This doesn't matter when you are dealing with basic arithmetic because intuition is the best guide there, but it becomes critical when you are dealing with concepts involving meta-arguments with several levels of abstraction, and intuition is nowhere to be found. The problem with only understanding arithmetic at an intuitive level is that you then have no basis for extending it to more abstract concepts. For instance, how will you know what constitutes a valid mathematical proof (I sometimes find myself wondering that myself when reading a particularly abstract or difficult proof)?

The point of the Principia was to show that all mathematical objects are just properties of properties (the way Frege had it, for instance, the number 0 is just the property of being a property whose extension is in one-one correspondence with that of the property of not being self-identical). On this view, all mathematical theorems become theorems about logic. This project is part of early proof-theory, which is now used to verify very complicated mathematical arguments, ones which are too difficult to check by hand.

epepke wrote:I haven't read Wittgenstein's critique and only portions of the Principia Mathematica. However, when people talk about counting, I wonder things like how many eggs is a double-yolked egg? This is similar to other ideas like how many photons are there, or how many electrons there are within a 95% electron density surface. Also, I usually have a half an onion around the house which is more onion than some whole onions. I'm not sure that counting is as basic an idea as it's made out to be.

I've read second-hand about cultures that only count up to 8 or haven't made the abstraction from "one apple", "one arrow", "one person" to *one*.

The idea that numbers are something like Frege cardinals was sufficiently popular at the time to work its way into at least one just-so story of how cultures invent numbers.

The process of counting is an attempt to abstract the commonality of sets of differing elements that have had their separate elements put into one-to-one correspondence with elements of other sets and determining that all those sets have no elements more and no elements less than the other sets.
To determine that commonality we have devised the set of counting numbers and we bring each element of a set into one-to-one correspondence with the set of counting numbers in a well defined and universally accepted and understood order and the last number of the count is the number of the set. The number of the set is the commonality of the chosen sets.
For example a set of 5 tortoises and a set of 5 chocolate bars share their fiveness if nothing else.
What is interesting is to recognize that the cardinality/ number of the set is, at first, the sound of each number that we make as we count and that is the first degree of abstraction, the sound can be represented as a word , eg one, two three etc, a second degree of abstraction, and the words can be embodied in a symbol 1,2, 3 etc a third degree of abstraction and when we manipulate those numbers in our heads eg adding 2 and 2 we have a fourth degree of abstraction.
We count aloud in many different languages different sounds and words but in general we do the penultimate abstraction in the representing of the number in Arabic numerals using a decimal/ denary number base.
The apparently most simple aspect of mathematics, the concept of cardinality is profound.
What Wittengenstein is saying , I think, is that if an error occurs in counting the error is with the one counting and not the process and also if the Principia shows the counting process as wrong then the Principia is in error.
I am not well versed in this but I think Godel went on to show that the Principia was incomplete and could never be complete when discussing the Natural/ counting numbers.