Parsing the Principia Mathematica

[xwolfhunter]

I know that this book is both fundamentally flawed and an extremely important book. I got it for Christmas, and I kind of view it like a whetstone for my mind. I'm taking my time and I won't turn the page unless I understand everything beforehand. I've been on the same page for a little bit, though, and I want someone who is familiar with it to help me out a little. I got the rough draft version, so I think it's partly that, but anyway.

I'm in the introduction, at the point where he discusses propositional functions, and he says, "Let $\phi x$ be a statement containing $x$ and such that it becomes a proposition when $x$ is given any fixed determined meaning. Then $\phi x$ is called a "propositional function"; it is not a proposition, since . . . it really makes no assertion at all." (Due to the ambiguity of $x$). Later on the same page he says "When we wish to speak of the propositional function corresponding to "$x$ is hurt," [in other words, $\phi x$, which is, according to him . . . a propositional function . . .] we shall write "$\hat{x}$ is hurt." Thus "$\hat{x}$ is hurt" is the propositional function and "$x$ is hurt" is an ambiguous value of that function."

Ignoring the contradiction in definition, the structure as I see it is we have some variable $x$, which can take the form of anything, potentially, but is never in and of itself a proposition. Then we have $\phi$, the functional variable, which is essentially a proposition that describes nothing. When we apply this to $x$, we get $\phi x$, which is the "propositionization" of $x$, and the completion of $\phi$, giving it something to describe. But still at this point, $\phi x$ is not a proposition, since it speaks of nothing in particular, though a specific member of nothing in particular. So far everything is in terms of a variable $x$. Now from the functional variable's side, saying nothing of $x$, we can define $\phi \hat{x}$ as the collection of all possible values of $x$ applied to $\phi$. $x$ itself has absolutely nothing to do with $\phi \hat{x}$, because $\hat{x}$ is an apparent variable, while $x$ is a free variable - right? Or . . . do we say $\phi \hat{x}$ is a propositional function, and $\phi x$ is a propositional function of $x$? Also, what exactly is $\phi \hat{x}$? It's not a proposition, but I don't think it's a variable proposition either, since it . . . has . . . no . . . variable? We can't say $\phi \hat{x}$ is true, because $(x) \cdot \phi x$ is what we'd use to . . . or is $\phi \hat{x}$ always true regardless . . .

I think I'm like right on the verge of exact comprehension here, but the book is not written for the layperson, which I very much am, so a little elucidation would be very much appreciated.

 

[PeroK]

I know that this book is both fundamentally flawed and an extremely important book. I got it for Christmas, and I kind of view it like a whetstone for my mind. I'm taking my time and I won't turn the page unless I understand everything beforehand. I've been on the same page for a little bit, though, and I want someone who is familiar with it to help me out a little. I got the rough draft version, so I think it's partly that, but anyway.

I'm in the introduction, at the point where he discusses propositional functions, and he says, "Let $\phi x$ be a statement containing $x$ and such that it becomes a proposition when $x$ is given any fixed determined meaning. Then $\phi x$ is called a "propositional function"; it is not a proposition, since . . . it really makes no assertion at all." (Due to the ambiguity of $x$). Later on the same page he says "When we wish to speak of the propositional function corresponding to "$x$ is hurt," [in other words, $\phi x$, which is, according to him . . . a propositional function . . .] we shall write "$\hat{x}$ is hurt." Thus "$\hat{x}$ is hurt" is the propositional function and "$x$ is hurt" is an ambiguous value of that function."

Ignoring the contradiction in definition, the structure as I see it is we have some variable $x$, which can take the form of anything, potentially, but is never in and of itself a proposition. Then we have $\phi$, the functional variable, which is essentially a proposition that describes nothing. When we apply this to $x$, we get $\phi x$, which is the "propositionization" of $x$, and the completion of $\phi$, giving it something to describe. But still at this point, $\phi x$ is not a proposition, since it speaks of nothing in particular, though a specific member of nothing in particular. So far everything is in terms of a variable $x$. Now from the functional variable's side, saying nothing of $x$, we can define $\phi \hat{x}$ as the collection of all possible values of $x$ applied to $\phi$. $x$ itself has absolutely nothing to do with $\phi \hat{x}$, because $\hat{x}$ is an apparent variable, while $x$ is a free variable - right? Or . . . do we say $\phi \hat{x}$ is a propositional function, and $\phi x$ is a propositional function of $x$? Also, what exactly is $\phi \hat{x}$? It's not a proposition, but I don't think it's a variable proposition either, since it . . . has . . . no . . . variable? We can't say $\phi \hat{x}$ is true, because $(x) \cdot \phi x$ is what we'd use to . . . or is $\phi \hat{x}$ always true regardless . . .

I think I'm like right on the verge of exact comprehension here, but the book is not written for the layperson, which I very much am, so a little elucidation would be very much appreciated.

I'm a little out of my depth here, but anyway. What I think he's trying to say is that there is a propositional function that exists separately from the proposition as applied to a variable (ambiguous value) and, indeed, as applied to a specific value of that variable. And there's a need to establish a notational convention to distinguish the function from the function evaluated at a variable.

This is an issue in mathematics as well, as we talk about the function $f$ and then $f(x)$ is the function evaluated at a variable and $f(0)$ is the function evaluated at a specific value. That's fine.

But, it can be quite difficult to avoid using $f(x)$. For example, you can talk about the $sin$ function without an argument, but what about $3sin^2(5x)$? How do you reference that function without using function values? How do you emphasise that it's the function itself and not a function value that you mean?

One idea, which I think is analogous to the idea in the book, is to write the function as something like $3sin^2(5X)$ which differentiates the function itself from the function value at a given $x$.

I should add that I don't see any contradiction in what he's saying.

 

[xwolfhunter]

I'm a little out of my depth here, but anyway. What I think he's trying to say is that there is a propositional function that exists separately from the proposition as applied to a variable (ambiguous value) and, indeed, as applied to a specific value of that variable. And there's a need to establish a notational convention to distinguish the function from the function evaluated at a variable.

This is an issue in mathematics as well, as we talk about the function $f$ and then $f(x)$ is the function evaluated at a variable and $f(0)$ is the function evaluated at a specific value. That's fine.

But, it can be quite difficult to avoid using $f(x)$. For example, you can talk about the $sin$ function without an argument, but what about $3sin^2(5x)$? How do you reference that function without using function values? How do you emphasise that it's the function itself and not a function value that you mean?

One idea, which I think is analogous to the idea in the book, is to write the function as something like $3sin^2(5X)$ which differentiates the function itself from the function value at a given $x$.

I should add that I don't see any contradiction in what he's saying.

Well it's not so much a contradiction . . . he explicitly says that $\phi x$ is a propositional function, and then explicitly says that the propositional function corresponding to $\phi x$ is $\phi \hat{x}$, and thereafter he says that anything of the form $\phi \hat{x}$ is a propositional function, while anything of the form $\phi x$ is an evaluation of that propositional function at $x$. It's like a misnomer, I guess.

That definitely helps, thank you. I see the relationship between $\phi \hat{x}$ and $\phi x$ clearly, but I still don't see where they fit into the structure of the book in general. $\phi x$ is not a proposition, so what is it? An argument? And is $\phi \hat{x}$ a proposition? What does $\hat{x}$ mean on its own?

 

[PeroK]

Well it's not so much a contradiction . . . he explicitly says that $\phi x$ is a propositional function, and then explicitly says that the propositional function corresponding to $\phi x$ is $\phi \hat{x}$, and thereafter he says that anything of the form $\phi \hat{x}$ is a propositional function, while anything of the form $\phi x$ is an evaluation of that propositional function at $x$. It's like a misnomer, I guess.

That definitely helps, thank you. I see the relationship between $\phi \hat{x}$ and $\phi x$ clearly, but I still don't see where they fit into the structure of the book in general. $\phi x$ is not a proposition, so what is it? An argument? And is $\phi \hat{x}$ a proposition? What does $\hat{x}$ mean on its own?

In my mathematical language I would say:

$\hat{x}$ is a dummy variable.
$\phi \hat{x}$ is a propositional function
$\phi x$ is the propositional function for an ambiguous/arbitrary/variable $x$
$\phi Obama$ is a proposition about the US President

When you say you don't see where this fits in, what do you imagine this book is about? It's about stripping logic and mathematics right down to the bare bones. If you are doing maths and, casually, your textbook says:

Let $f(x) > a$

To a mathematics student working "normally" it's quite simply a statement/proposition about an arbitrary/unknown function $f$ evaluated at an arbitrary element in its domain $x$ and an arbitrary constant $a$. But, to a logician, I imagine, it's a high-level abstract statement that requires a considerable logico-mathematical structure to support it.

You can do two things, now. You can accept that there is nothing more to be explained here (as I confess I do) and move on to calculus and linear algebra and study physics and relativity and quantum mechanics. Or, you can decide not to accept this and want to find the mathematical and logical bedrock on which you can build such a statement. It's up to you what road you take, but you need to be careful you don't descend into a bottomless pit!

 

[pbuk]

Or, you can decide not to accept this and want to find the mathematical and logical bedrock on which you can build such a statement. It's up to you what road you take, but you need to be careful you don't descend into a bottomless pit!

Note that the axioms of principia mathematica are NOT the bedrock on which modern mathematics is based, so study of its theorems is of historical interest only.

 

[Mark44]

You can do two things, now. You can accept that there is nothing more to be explained here (as I confess I do) and move on to calculus and linear algebra and study physics and relativity and quantum mechanics.

Note that the axioms of principia mathematica are NOT the bedrock on which modern mathematics is based, so study of its theorems is of historical interest only.

I agree completely with the above opinions. Speaking for myself, I have no interest in Whitehead & Russell's opus. Long ago someone told me that W & R burrowed so deeply into the foundations of logic and mathematics, that they weren't able to prove that 1 + 1 = 2 into well into the 3rd volume. I don't know for a fact that this is true, but whatever interest I might have had in this volume vanished when I heard that,

 

[Demystifier]

I know that this book is both fundamentally flawed and an extremely important book.

I don't want to be an advocate of my avatar (), but I wouldn't say that this book is "flawed". Or at least, not in the sense of being wrong. It's only that this approach turns out to be too complicated for practical purposes. Today we have a much simpler and more practical system called ZFC.

 

[xwolfhunter]

To a mathematics student working "normally" it's quite simply a statement/proposition about an arbitrary/unknown function $f$ evaluated at an arbitrary element in its domain $x$ and an arbitrary constant $a$. But, to a logician, I imagine, it's a high-level abstract statement that requires a considerable logico-mathematical structure to support it.

You can do two things, now. You can accept that there is nothing more to be explained here (as I confess I do) and move on to calculus and linear algebra and study physics and relativity and quantum mechanics. Or, you can decide not to accept this and want to find the mathematical and logical bedrock on which you can build such a statement. It's up to you what road you take, but you need to be careful you don't descend into a bottomless pit!

And now I see the difference between a logician and a mathematician! So I'm at a crossroads, I see.

Note that the axioms of principia mathematica are NOT the bedrock on which modern mathematics is based, so study of its theorems is of historical interest only.

Or, as is the case with me, as a whetstone for my mind. Also, I read that a recurring nightmare of Russell's was that the last copy of his Principia was, in the far future, tossed in the garbage as an unknown and unimportant text. Sad that this will eventually be its fate.

Long ago someone told me that W & R burrowed so deeply into the foundations of logic and mathematics, that they weren't able to prove that 1 + 1 = 2 into well into the 3rd volume. I don't know for a fact that this is true, but whatever interest I might have had in this volume vanished when I heard that,

Hmm . . . maybe I'll just read the first volume and move on to something more relevant. If I were interested in a book expounding ZFC, do you know of one you would recommend?

 

[Demystifier]

If I were interested in a book expounding ZFC, do you know of one you would recommend?

T. Jech, Set Theory is sort of bible for this.

 

[micromass]

Sad that this will eventually be its fate.

It's not sad at all. It's a horribly complicated work. It is a great accomplishment of mathematics that we were able to simplify this significantly. You should never be sad about a certain approach being replaced by a superior one. One should be sad about a superior approach being replaced by a worst one, something which does happen quite frequently in mathematics too.

The Principia is important for philosophy. For mathematicians, the book had had no significant influence. The only influence it could have had is in showing which approach does not work.

If you're interested in a development of math in the same spirit of the Principia, but then with modern set theory, there's Suppes' axiomatic set theory. I wouldn't really recommend this book if you're planning to be a mathematician though, there are better books then, but if you're into the philosophy of math then this book is ok.

 

[xwolfhunter]

It's not sad at all. It's a horribly complicated work. It is a great accomplishment of mathematics that we were able to simplify this significantly. You should never be sad about a certain approach being replaced by a superior one. One should be sad about a superior approach being replaced by a worst one, something which does happen quite frequently in mathematics too.

The Principia is important for philosophy. For mathematicians, the book had had no significant influence. The only influence it could have had is in showing which approach does not work.

If you're interested in a development of math in the same spirit of the Principia, but then with modern set theory, there's Suppes' axiomatic set theory. I wouldn't really recommend this book if you're planning to be a mathematician though, there are better books then, but if you're into the philosophy of math then this book is ok.

Sad that a work to which a man devoted many years of his life will ultimately be forgotten, I mean.

I can learn a whole heck of a lot from my rough draft copy of the Principia Mathematica, about logic and about thinking in general, and since that's what I have, I might as well do that. Thanks for the recommendation, I'm not sure at this point if I'll want to read another book on the foundations of mathematics or if I'll want to move on to something actually in the realm of pure mathematics when I'm finished with this book. We'll see.

 

[micromass]

Sad that a work to which a man devoted many years of his life will ultimately be forgotten, I mean.

That is true. But that's true for a lot of mathematics. If you study the history of mathematics, you'll see a lot of people spending so many years of their lifes on something that is now only studied in historic context. Russell-Whitehead is definitely not the only example of it, and it's not even the most extreme example since the book is still well-known 100 years later.

I can learn a whole heck of a lot from my rough draft copy of the Principia Mathematica, about logic and about thinking in general,

I kind of disagree that you'll learn much about logic and thinking from the Principia...

 

[xwolfhunter]

That is true. But that's true for a lot of mathematics. If you study the history of mathematics, you'll see a lot of people spending so many years of their lifes on something that is now only studied in historic context. Russell-Whitehead is definitely not the only example of it, and it's not even the most extreme example since the book is still well-known 100 years later.
I kind of disagree that you'll learn much about logic and thinking from the Principia...

So . . . you're saying that, if I read the whole of the Principia Mathematica, my mathematical visualization and clarity, my understanding of how the rules of logic are to be used, my familiarity with proofs, my working memory, etc., will be entirely unaffected? I would not argue (I would not be able to argue) that the Principia is not the worst math book ever written, but it is a math book, generally valid in its conclusions and extensive in its topics, deep and filled with a broad coloring by one of the greatest minds of the 20th century. That's what I see it as, and it is because of that sight that I am going to read it and learn a metric crap ton of things from it. I don't see how I could not learn a lot about logic and thinking from it. Have you read it carefully and can you actually firmly and empirically state the contrary? Or perhaps are you just biased?

Edit: I know, that was dickish of me, but do you have to rain on my parade here? I'm really enjoying this book so far, and so far it is teaching me. Perfect is the enemy of good, and I don't need an in-vogue math book to learn a lot about math.

 

[micromass]

So . . . you're saying that, if I read the whole of the Principia Mathematica, my mathematical visualization and clarity, my understanding of how the rules of logic are to be used, my familiarity with proofs, my working memory, etc., will be entirely unaffected?

As applicable to modern mathematics is done, yes: it will be almost entirely unaffected. As applicable to texts similar to the Principia (not that there are many), it will be very useful.

The Principia Mathematica is an extremely dangerous book to read if you're not very familiar with proofs and logic. It uses a kind of proof system that is never used outside of mathematics. If you attempt to read a math book or write a math paper, the kind of proofs of the Principia Mathematica will be horrible to do. A big part in learning math is learning how to read and write proofs. This is very important, and many students struggle significantly with it. Reading a book like the Principia Mathematica will definitely lead you astray since it teaches you the wrong way of reading and writing proofs. I speak from experience, I have guided many students and whenever they get this kind of introduction to proofs, it is very difficult to undo it and to set them straight. Their proofs in the beginning stages are simply horrible.

Your mathematical visualization will be unaffected, since there is no visualization in the Principia Mathematica.

 

[micromass]

Edit: I know, that was dickish of me, but do you have to rain on my parade here? I'm really enjoying this book so far, and so far it is teaching me. Perfect is the enemy of good, and I don't need an in-vogue math book to learn a lot about math.

OK sure, I'm sorry. Just go ahead reading the book then. If you enjoy the book, then there's no reason to stop. I just wanted to make sure you're doing it for the right reasons.

 

[xwolfhunter]

As applicable to modern mathematics is done, yes: it will be almost entirely unaffected. As applicable to texts similar to the Principia (not that there are many), it will be very useful.

The Principia Mathematica is an extremely dangerous book to read if you're not very familiar with proofs and logic. It uses a kind of proof system that is never used outside of mathematics. If you attempt to read a math book or write a math paper, the kind of proofs of the Principia Mathematica will be horrible to do. A big part in learning math is learning how to read and write proofs. This is very important, and many students struggle significantly with it. Reading a book like the Principia Mathematica will definitely lead you astray since it teaches you the wrong way of reading and writing proofs. I speak from experience, I have guided many students and whenever they get this kind of introduction to proofs, it is very difficult to undo it and to set them straight. Their proofs in the beginning stages are simply horrible.

Your mathematical visualization will be unaffected, since there is no visualization in the Principia Mathematica.

Then I'll simply use it to clear the cobwebs from my mind. I would imagine that if mathematicians can do math in multiple fields of mathematics, it will not be so much of a stretch to assume that I can do the same, and if the text is really so absolutely useless, then in three years there will be hardly a shred of it left in my mind. I'm not too worried about it, the brain is defined by its neuroplasticity.

 

[xwolfhunter]

OK sure, I'm sorry. Just go ahead reading the book then. If you enjoy the book, then there's no reason to stop. I just wanted to make sure you're doing it for the right reasons.

I definitely appreciate that, I have a new context in which to view the book as I go along. If the book becomes boring, I won't think twice before I switch it up. Thanks.