When you see X ⊃ Y, you are to read "X implies Y" or "If X, then Y." Sometimes you may see the modern form, X —> Y, in my comments.
When you see X ≡ Y, you should read "X equivalent to Y" or "X strictly implies Y" or "If and only if X, then Y" or "If X, then Y and if Y, then X" (these forms are sometimes shortened to "iff"). "X ≡ Y" can also be read: "Y is necessary for X and X is sufficient for Y." Occasionally one also sees "X ≡ Y" interpreted as "Y holds just in case X holds" or "Y holds only in case X holds" and the converse, "X holds just in case Y holds." Sometimes my comments use the modern X <—> Y.
For the quantifier all, you will see ( ), as in (x)Fx. In my comments I sometimes use the modern symbol ∀, as in ∀xFx. The existential quantifer ∃, as in ∃xFx, is used in W's text and in my comments. P●Q should be read "P and Q." Often the dot is dropped, as in PQ. Occasionally dots are used instead of parentheses.
In a tart response to a critic, Bertrand Russell once pointed out that he no longer held the philosophical views in question.
Yet, the Library of Living Philosophers printed the critic's analysis anyway. The editor of that series, Paul Arthur Schilpp, was following the time-honored rule that even disavowed philosophical positions could be worthy of discussion.
Well-known is the fact that Ludwig Wittgenstein later changed his mind about Tractatus, having been convinced by Frank Ramsay that it did not, after all, solve all major questions. Wittgenstein then went on to a philosophy that is neatly summed by Marshall Macluhan's aphorism, "the medium is the message."
Tractatus, with its quasi-engineering format and style, made a powerful impact on the "hard-headed realists," who saw it as an antidote to wooly-minded metaphysics and just what the doctor ordered as a backup for 20th Century scientific materalism or materialist science – though three years after its publication the quantum mechanics revolution would throw a monkey wrench into the whole materialist agenda. Nevertheless, the "logical positivists" – who counted Wittgenstein as much as Comte among their forebears – carried the banner of "scientific" philosophy for much of the century.
Yes, it is true that logical positivism no longer is in vogue. Yet, it will be conceded that the "system" of logical positivism underlies the unspoken assumptions of a great many people among the educated and semi-educated classes.
To me, that alone provides adequate reason to revisit Tractatus with a view to questioning its assertions and assumptions.
Clearly the brilliance of Wittgenstein's contribution, flawed though it is, makes it worthy of discussion even today. Yet we must concede that the century since publication has seen an enormous proliferation of work in mathematical logic (sometimes called logistic) and set theory, thus making some of these comments passe. Hopefully, not all.
Though, as the Bradley-Russell debate shows, relations constitute an imperfect vehicle for representing human reality, they are nevertheless integral to human communication. I would even go so far as to define mathematics as "the science of exact binary relations." (Rxyz can be put into binary terms, as in the ordered pairs < x, < y,z > >.)
I regard a relation as a verb-like construct that pairs two noun-like constructs. The noun can represent an abstract picture of some ponderable object or of an "action." An action is abstracted from a number of specific events. For example, there is no such thing as running devoid of runners. We may say that runs relates some object, usually some animal, to something else.
But on that ground, one might argue that Joline runs doesn't seem to be a relation.
In such a case, we have two options: We may say that for aRb (or R〈a,b〉) , b = ∅. Or we may note that Joline runs abbreviates the picture Joline moves (by running) her body.
And we must concede that there is nothing to stop an infinite regress type of relation, as in
R〈a, 〈 b,〈 c, 〈 ... 〉 〉 〉 〉 etc. Or, we can write aRb, b=cSd, d=eTf... and so on. In fact, such relations are typical in mathematics. The philosophical problem arises in trying to define a relation in terms of another relation.
That problem, in fact, has much of Russell's paradox about it, at least insofar as the self-referencing aspect is concerned.
Then there is aR(aRb), which also brings up the self-referencing issue.
Despite these cautions, the way we ordinarily communicate with each other is well expressed as a system of relations, both empirical and logical/formal.
We may take every word in the Oxford dictionary as the set A and form the pairings A X A. We then take every word in that dictionary that has even a vague possibility of representing some action-type notion and call that a relation.
Among subsets of relations are R〈 a,k〉 and R〈k,b〉 where a and b represent variables and k is held constant.
All such relations represent propositions or assertions.
Empirical assertions are judged true or false according to whether we believe they accord with a purported objective reality.
Logical propositions follow rules laid down by logicians that reflect commonly accepted means of human reasoning. Such formal propositions are true or false based on their internal consistency. AB true requires that both A and B be adjudged true, and that this judgment must stem from axioms (though even that proviso is not altogether so, as Goedel showed). A~A is false by almost universal agreement.
I don't wish to rehearse all the arcana of the theory of logic here. My point is that the notion of sets of relations covers the field of logic very well.
Well, yes, it is true that many relation pairs are gibberish – at present – though someone may pair two words/images/ideas in some technical or poetic way that would be meaningful to a subset of hearers/viewers. Thus, we may say that most if not all pairs and their relation potentially are meaningful. By meaningful, we are saying that a binary truth value is applicable to the specific relation/proposition.†
But it follows that only some small subset C of A X B would represent meaningful propositions. That is, if we regard p as the specific relation R〈 a,b〉, where here a and b are each instances, we find that p is a member of A X A, where A is the set of all words and concept symbols used in English. A proposition however is regarded as carrying a truth value. Most members of A2 are not regarded by anyone as meaningful relation pairs and so are not propositions, though in future any non-proposition may take on a sense that can be affirmed or denied, thus becoming a proposition.
Denial can only be applied to members of the set of propositional relations P.
That is, ~p is a member of Pc, or ~p ∈ Pc <—> p ∈ P. Pc is not a subset of A2. Pc is a subset of (A2)c.
In a sense, can we not say that the formalism of relations and the generally accepted rules of logic show without further ado that the human brain/mind is wired to a primary in-built grammar. What human language cannot be cast in the form of ordered pairs of relation sets? This is not to say that the work of Chomsky et al is pointless, but to say that it seems that that endeavor was long prefigured by the work of the logicians and philosophers.
Russell did not like Bradley's objection that while relations have a practical value, they cannot exist intrinsically. He favored the idea that any infinite regress was harmless –> though he certainly did not feel that way about his own epynonymous paradox.
My view is that, as the relations system fully encompasses all of ordinary logic (including mathematical logic), Wittgenstein's attempt to define away Russell's bugbear cannot be held to have succeeded. That is, the edifice of logic contains the very sort of self-referencing difficulty elucidated by Russell for set theory and by Goedel for formal systems in general.
This further suggests that humans can not "know" the truths of their existence based on logic alone.
As the set of propositions B is a subset of A X A, it follows that B is finite. This means that each proposition can be numbered, reducing it to being represented without quantifier or variable. But, it is more convenient to write ∃xFx, which represents a subset F< a, x >, where F and a are held constant and x varies, or F< x, a>. Multiple relations can be expressed as pairs, of course. a < b < c is the less-than relation L, written L(a, (b,c)) [Here I use the rounded parenthetical mark in order to avoid confusion between "<" for "less than" and "< , >" for "ordered pair set."]
On the matter of the validity of relations as a means of discussing verbalization, I would point out that we really have no alternative. That's how most of us speak/think: via relations. That is not to say that idealists don't have a point when they deny that validity. In other words, it may well be that relations do not yield a proper conception of reality. Yet, in our day-to-day affairs, they work well enough.
Hence the dispute over "internal" versus "external" relations might be seen in that light. W, though not an idealist, seems to be aware of the concern about these distinctions.
A.C. Ewing, in Idealism, A Critical Survey (Methuen 1933), writes that the controversy arose from the idealist belief that "all reality is an interconnected system such that a difference in one part would always involve a difference in others."‡
Now, concerning the relation "equals," Tractatus, as others, take that to mean "same as." That can be a definition.
But consider the relation set =<2+2,4>, or that is, "2+2 = 4". Crudely, we teach children that "2 and 2 is the same as 4." But that is an abbreviated way of saying something more complex:
One way of expressing this would be
Thus I suggest that "=" be taken as a special case of the equivalence relation "≡". I.e. "2 ∪ 2' ≡ 4" in this case means "2 + 2 = 4".
As Tarski pointed out "truth" is an indefinable term in mathematics. But some form of the concept is still necessary, whether it be called "satisfiability" or some such. In our view of relations, by "truth" we imply that relation R poses a yes or no question for the pair x and y: "Does x 'go with' y in the order < x,y >?" Or is there a reason to say "x certainly does not belong with y in the specified order"? E.g., "Does 2+2 go with 4 in the equals relation?" Yes. "Does 2+1 go with 4 in the equals relation"? No. [Here R is symmetric.] That is, < 2+1,4 > ∈ Rc where R ⊂ N X N (and any sum n+m ∈ N).
Moreover, one can see how our example draws together the correspondence and coherence theories of truth. That is, we say xRy corresponds to a truth value of T "because" x goes with y under R. Yet, we cannot blindly say that that is so. First, we have to justify that claim with a set of other relation/assertions. Of course, we cannot insist that all truths derive from a few interdependent axioms, as perhaps the idealists might say. In a pluralist cosmos, the axioms would not necessarily be interdependent. Also, as a practical matter, we can only assess parts of "reality" and cannot really find a perfect set of propositional relations that reflects a purported Theory of Everything. In other words, perhaps the coherence theory is valid, but we for the most part must make do with the correspondence notion.
In any case, a modern form of the axiom is, as Wikipedia reports:
But rather than taking a positive approach to saying what a set can be, it seems to me better to say what it cannot be, So we require an axiom schema that says
By formalizing what cannot be a set in this way, we don't need to axiomatize the concept of N The successor algorithm given above tells us that any natural number may be so defined. We give the null set, ∅, the name 0. 1 is defined as ∅ ∪ {∅}, which reduces to {∅}. 2 is the name given to {∅} ∪ {{∅}} = {{∅},{{∅}}}. You can see that any natural number will follow upon iteration.
We do not say axiomatically that there is an infinity. We say that there is a set N that contains all the naturals defined as shown. The set N is the name given for the entire collection. N is barred from being a member of N (that claim rests on a different axiom though if we define N that way, we can in this instance bypass that axiom also). We then define finite as a quality of any member of N. Any set that is bijective with an n is finite. Hence, N is infinite. (The quality is that n —> 1+n, but N -/-> 1+N.)
This formulation is found in NBG set theory, for example. In any case, we may argue that infinity no longer is axiomatic. Infinity is defined as a non-finite set. Finite is held to be descriptive of the cardinal for any member of N.
Another standard way to define infinity set theoretically:
Of course, the inductive definition above dovetails with the subset definition.
Rosser's axiom scheme 13, along with each statement obtained from it by prefixing some set of universal quantifiers, is given as:
In fact, I suggest that, despite the proof (by Spector) that the axiom could be derived from other NF axioms, philosophically it was unnecessary in Rosser's approach. That is because Rosser in that book makes a point of distinguishing between "all" and "any." Of course, usually the truth value of a proposition that reads (all x)Px or (any x)Px is the same in either case, which is why we use only one universal quantifier ∀ . Yet, it is possible to define a set using an "any" specifier: α, as in αn(n --> n+1). We then give this aggregate the name N and observe that this α form defines the aggregate by intensionally specifying its elements, and does not violate the "hierarchy axiom" cited above. Hence, it is a set. (Before we know that a particular aggregate is a set, we say that x is not necessarily a member of the aggregate but only an affiliate of it.) Of course, one may suppose that that axiom appears to imply infinity. But we may apply the α modifier to that axiom, without requiring an understanding of completion or closure. That is, the α modifier may be read to imply closure, but it need not be read that way.
In any case, these observations, along with Rosser's unique approach to logic in his Logic for Mathematicians (1953), reminds one that there are many roads in formal logic, not one – though of course the basics are, for the most part, unchanging.
In addition, I suppose it might be mentioned that the status of Tractatus has suffered considerable decline among mathematical logicians. Kurt Gödel noted that he had given it only "superficial" attention, though he was involved in the Vienna Circle as a young man. He minimized W's influence on that philosophical group.
By 1983, an important anthology of mathematical philosophy had dropped Tractatus from its second edition in favor of the work of others.* It seems, then, that despite Russell's sponsorship, Tractatus was more influential among those logical positivists who were not primarily mathematicians or mathematical logicians. An important exception was Rudolf Carnap,†† whose notion that logical syntax could replace outworn philosophy echoes the thrust of Tractatus. In fact, Carnap focused the Vienna Circle on Tractatus in 1926 and 1927, according to Hao Wang.‡‡ Gödel, in fact, suppressed a detailed review he wrote of Carnap's philosophy for the Library of Living Philosophers. Carnap's attempt to carry out what he took to be W's vision is not generally accepted as having succeeded.
In any case, I don't wish to undervalue Tractatus, which was thought out by a young man in a prisoner of war camp. As I mentioned, the aim here is to throw a spotlight on one of the supposed philosophical pillars of the modern educated elite, most of whom of course have only the vaguest inkling of its import.
So then, how firm is the foundation on which they stand?
A positive summary
of Tractatus and W's other work is found at The School of Life, a YouTube channel:
https://www.youtube.com/watch?v=pQ33gAyhg2c
Freeman Dyson,
the physicist, relates a less-than-heartwarming encounter with Wittgenstein.
https://www.youtube.com/watch?v=byi3vOnVodQ
Dyson calls W a "charlatan" who liked to torment people.
Argues that W slipped up on Goedel theorem
http://wab.uib.no/agora/tools/alws/collection-6-issue-1-article-6.annotate
Kurt Gödel thought
W's later philosophy had "taken a step backward" from Tractatus, according to Hao Wang in Reflections on Kurt Gödel (MIT 1987). Wang adds that in 1972 Gödel noted that he had read Tractatus in 1927, though "never thoroughly."
†. In probability theory, the truth value system ought remain binary. Middle values pertain to degree of uncertainty. There is a case for a third truth value of 0, for provably undecidable. But, a problem with barring the logical rule of excluding the middle is that it eviscerates whole sections of mathematics that mathematicians are loath to give up.
Footnotes
++. The subset axiom is the axiom of Zermelo-Fraenkel set theory which asserts the existence for any set a and a formula A(y) of a set x consisting of all elements of a satisfying A(y).
This axiom was introduced by Zermelo.
Points of interest:
‡ . Current physics denies that this can always hold because of the claim that some parts of the cosmos are too far away and moving too fast relative to us to ever be able to interact with our sector in any manner.
* . Philosophy of Mathematics: Selected Readings edited by Paul Benacerraf and Hilary Putnam (2d ed. Cambridge 1983; 1st ed. Prentice-Hall 1964).
† † . Reflections on Kurt Gödel by Hao Wang (MIT 1987).
‡‡ . According to André Carus's article on Carnap at the Stanford Encyclopedia of Philosophy (2020):
Yet, the Library of Living Philosophers printed the critic's analysis anyway. The editor of that series, Paul Arthur Schilpp, was following the time-honored rule that even disavowed philosophical positions could be worthy of discussion.
Well-known is the fact that Ludwig Wittgenstein later changed his mind about Tractatus, having been convinced by Frank Ramsay that it did not, after all, solve all major questions. Wittgenstein then went on to a philosophy that is neatly summed by Marshall Macluhan's aphorism, "the medium is the message."
Tractatus, with its quasi-engineering format and style, made a powerful impact on the "hard-headed realists," who saw it as an antidote to wooly-minded metaphysics and just what the doctor ordered as a backup for 20th Century scientific materalism or materialist science – though three years after its publication the quantum mechanics revolution would throw a monkey wrench into the whole materialist agenda. Nevertheless, the "logical positivists" – who counted Wittgenstein as much as Comte among their forebears – carried the banner of "scientific" philosophy for much of the century.
Yes, it is true that logical positivism no longer is in vogue. Yet, it will be conceded that the "system" of logical positivism underlies the unspoken assumptions of a great many people among the educated and semi-educated classes.
To me, that alone provides adequate reason to revisit Tractatus with a view to questioning its assertions and assumptions.
Clearly the brilliance of Wittgenstein's contribution, flawed though it is, makes it worthy of discussion even today. Yet we must concede that the century since publication has seen an enormous proliferation of work in mathematical logic (sometimes called logistic) and set theory, thus making some of these comments passe. Hopefully, not all.
A few remarks on relations
[This section to be polished up later.]Though, as the Bradley-Russell debate shows, relations constitute an imperfect vehicle for representing human reality, they are nevertheless integral to human communication. I would even go so far as to define mathematics as "the science of exact binary relations." (Rxyz can be put into binary terms, as in the ordered pairs < x, < y,z > >.)
I regard a relation as a verb-like construct that pairs two noun-like constructs. The noun can represent an abstract picture of some ponderable object or of an "action." An action is abstracted from a number of specific events. For example, there is no such thing as running devoid of runners. We may say that runs relates some object, usually some animal, to something else.
But on that ground, one might argue that Joline runs doesn't seem to be a relation.
In such a case, we have two options: We may say that for aRb (or R〈a,b〉) , b = ∅. Or we may note that Joline runs abbreviates the picture Joline moves (by running) her body.
And we must concede that there is nothing to stop an infinite regress type of relation, as in
R〈a, 〈 b,〈 c, 〈 ... 〉 〉 〉 〉 etc. Or, we can write aRb, b=cSd, d=eTf... and so on. In fact, such relations are typical in mathematics. The philosophical problem arises in trying to define a relation in terms of another relation.
That problem, in fact, has much of Russell's paradox about it, at least insofar as the self-referencing aspect is concerned.
Then there is aR(aRb), which also brings up the self-referencing issue.
Despite these cautions, the way we ordinarily communicate with each other is well expressed as a system of relations, both empirical and logical/formal.
We may take every word in the Oxford dictionary as the set A and form the pairings A X A. We then take every word in that dictionary that has even a vague possibility of representing some action-type notion and call that a relation.
Among subsets of relations are R〈 a,k〉 and R〈k,b〉 where a and b represent variables and k is held constant.
All such relations represent propositions or assertions.
Empirical assertions are judged true or false according to whether we believe they accord with a purported objective reality.
Logical propositions follow rules laid down by logicians that reflect commonly accepted means of human reasoning. Such formal propositions are true or false based on their internal consistency. AB true requires that both A and B be adjudged true, and that this judgment must stem from axioms (though even that proviso is not altogether so, as Goedel showed). A~A is false by almost universal agreement.
I don't wish to rehearse all the arcana of the theory of logic here. My point is that the notion of sets of relations covers the field of logic very well.
Well, yes, it is true that many relation pairs are gibberish – at present – though someone may pair two words/images/ideas in some technical or poetic way that would be meaningful to a subset of hearers/viewers. Thus, we may say that most if not all pairs and their relation potentially are meaningful. By meaningful, we are saying that a binary truth value is applicable to the specific relation/proposition.†
But it follows that only some small subset C of A X B would represent meaningful propositions. That is, if we regard p as the specific relation R〈 a,b〉, where here a and b are each instances, we find that p is a member of A X A, where A is the set of all words and concept symbols used in English. A proposition however is regarded as carrying a truth value. Most members of A2 are not regarded by anyone as meaningful relation pairs and so are not propositions, though in future any non-proposition may take on a sense that can be affirmed or denied, thus becoming a proposition.
Denial can only be applied to members of the set of propositional relations P.
That is, ~p is a member of Pc, or ~p ∈ Pc <—> p ∈ P. Pc is not a subset of A2. Pc is a subset of (A2)c.
In a sense, can we not say that the formalism of relations and the generally accepted rules of logic show without further ado that the human brain/mind is wired to a primary in-built grammar. What human language cannot be cast in the form of ordered pairs of relation sets? This is not to say that the work of Chomsky et al is pointless, but to say that it seems that that endeavor was long prefigured by the work of the logicians and philosophers.
Russell did not like Bradley's objection that while relations have a practical value, they cannot exist intrinsically. He favored the idea that any infinite regress was harmless –> though he certainly did not feel that way about his own epynonymous paradox.
My view is that, as the relations system fully encompasses all of ordinary logic (including mathematical logic), Wittgenstein's attempt to define away Russell's bugbear cannot be held to have succeeded. That is, the edifice of logic contains the very sort of self-referencing difficulty elucidated by Russell for set theory and by Goedel for formal systems in general.
This further suggests that humans can not "know" the truths of their existence based on logic alone.
As the set of propositions B is a subset of A X A, it follows that B is finite. This means that each proposition can be numbered, reducing it to being represented without quantifier or variable. But, it is more convenient to write ∃xFx, which represents a subset F< a, x >, where F and a are held constant and x varies, or F< x, a>. Multiple relations can be expressed as pairs, of course. a < b < c is the less-than relation L, written L(a, (b,c)) [Here I use the rounded parenthetical mark in order to avoid confusion between "<" for "less than" and "< , >" for "ordered pair set."]
On the matter of the validity of relations as a means of discussing verbalization, I would point out that we really have no alternative. That's how most of us speak/think: via relations. That is not to say that idealists don't have a point when they deny that validity. In other words, it may well be that relations do not yield a proper conception of reality. Yet, in our day-to-day affairs, they work well enough.
Hence the dispute over "internal" versus "external" relations might be seen in that light. W, though not an idealist, seems to be aware of the concern about these distinctions.
A.C. Ewing, in Idealism, A Critical Survey (Methuen 1933), writes that the controversy arose from the idealist belief that "all reality is an interconnected system such that a difference in one part would always involve a difference in others."‡
Now, concerning the relation "equals," Tractatus, as others, take that to mean "same as." That can be a definition.
But consider the relation set =<2+2,4>, or that is, "2+2 = 4". Crudely, we teach children that "2 and 2 is the same as 4." But that is an abbreviated way of saying something more complex:
One way of expressing this would be
2 = 2.So the relation "=" here indicates more than "same as." It indicates a procedure which must be followed (at least inferentially) in order to match the left side of the relation with the right.
This says that the set named 2 is bijective with itself (which is not quite saying: "the same as itself.")
The "+" operation requires that either the set named m or the set named n must be used to form a set with which it is bijective. So m X ∅ will do, producing elements of
form < x, ∅ >. Calling that set m', we now make m' ∪ n, which (not proved here) is bijective with some x ∈ N. That x is bijective with the sum "m + n," which is the set m' ∪ n.
That is, 2 + 2 --> 2 ∪ 2', which is bijective with 4.
Thus I suggest that "=" be taken as a special case of the equivalence relation "≡". I.e. "2 ∪ 2' ≡ 4" in this case means "2 + 2 = 4".
As Tarski pointed out "truth" is an indefinable term in mathematics. But some form of the concept is still necessary, whether it be called "satisfiability" or some such. In our view of relations, by "truth" we imply that relation R poses a yes or no question for the pair x and y: "Does x 'go with' y in the order < x,y >?" Or is there a reason to say "x certainly does not belong with y in the specified order"? E.g., "Does 2+2 go with 4 in the equals relation?" Yes. "Does 2+1 go with 4 in the equals relation"? No. [Here R is symmetric.] That is, < 2+1,4 > ∈ Rc where R ⊂ N X N (and any sum n+m ∈ N).
Moreover, one can see how our example draws together the correspondence and coherence theories of truth. That is, we say xRy corresponds to a truth value of T "because" x goes with y under R. Yet, we cannot blindly say that that is so. First, we have to justify that claim with a set of other relation/assertions. Of course, we cannot insist that all truths derive from a few interdependent axioms, as perhaps the idealists might say. In a pluralist cosmos, the axioms would not necessarily be interdependent. Also, as a practical matter, we can only assess parts of "reality" and cannot really find a perfect set of propositional relations that reflects a purported Theory of Everything. In other words, perhaps the coherence theory is valid, but we for the most part must make do with the correspondence notion.
A word on the axiom of infinity
W never brings up this axiom in Tractatus but it seems to me that the whole thrust of his critique demands that he say something about it. He does bring up infinity, without acknowledging the philosophical problem posed by the concept, though his aim is to solve or eject all such problems through correct reasoning and language.In any case, a modern form of the axiom is, as Wikipedia reports:
∃N(∅ ∈ N ● ∀n ∈ N((n ∪ {n}) ∈ N)Wikipedia says that the set N is presumed infinite. Zermelo's original axiom, as I understand it, actually says that N, which designates the natural numbers, "is a set." Zermelo wished to avoid the paradoxes by tightening the definition of set, which could not now be any aggregate of objects, as defined intensionally. Otherwise one has, say, X = {x|x ∉ x}, which leads to is X ∈ X? If yes, no. If no, yes.
But rather than taking a positive approach to saying what a set can be, it seems to me better to say what it cannot be, So we require an axiom schema that says
~(x ∈ y)(y ∈ z)...(t ∈ x).The dots mean any string of (ui ∈ vi) of finite length. In words, "if x is in y, then no set that y is in may be in x." A picture is a sequence of nested containers. The largest box cannot be in any smaller one.
By formalizing what cannot be a set in this way, we don't need to axiomatize the concept of N The successor algorithm given above tells us that any natural number may be so defined. We give the null set, ∅, the name 0. 1 is defined as ∅ ∪ {∅}, which reduces to {∅}. 2 is the name given to {∅} ∪ {{∅}} = {{∅},{{∅}}}. You can see that any natural number will follow upon iteration.
We do not say axiomatically that there is an infinity. We say that there is a set N that contains all the naturals defined as shown. The set N is the name given for the entire collection. N is barred from being a member of N (that claim rests on a different axiom though if we define N that way, we can in this instance bypass that axiom also). We then define finite as a quality of any member of N. Any set that is bijective with an n is finite. Hence, N is infinite. (The quality is that n —> 1+n, but N -/-> 1+N.)
This formulation is found in NBG set theory, for example. In any case, we may argue that infinity no longer is axiomatic. Infinity is defined as a non-finite set. Finite is held to be descriptive of the cardinal for any member of N.
Another standard way to define infinity set theoretically:
If A is a proper subset of B and A and B are bijective, then B (and A) is infinite.Is an axiom needed? Isn't the subset axiom++ sufficient?
Of course, the inductive definition above dovetails with the subset definition.
N = {n|n --> n+1}, where n is defined recursively.I recall Barkeley Rosser making sure to tack onto his book on logic an appendix with a new proof that the axiom of infinity could be derived from other axioms, though I haven't had time to decode the proof. His axiom differs from others in that his book incorporates W.V. Quine's New Foundations approach to logic.
X = {n2|n2 --> (n+1)2}
Since X ⊂ N and since for every n there is exactly one n2, N must be infinite.
Rosser's axiom scheme 13, along with each statement obtained from it by prefixing some set of universal quantifiers, is given as:
∀(m,n) m,n ∈ N ● m+1 = n+1 ● --> ● m = nIn general, he refers to this statement as the "axiom of infinity."
In fact, I suggest that, despite the proof (by Spector) that the axiom could be derived from other NF axioms, philosophically it was unnecessary in Rosser's approach. That is because Rosser in that book makes a point of distinguishing between "all" and "any." Of course, usually the truth value of a proposition that reads (all x)Px or (any x)Px is the same in either case, which is why we use only one universal quantifier ∀ . Yet, it is possible to define a set using an "any" specifier: α, as in αn(n --> n+1). We then give this aggregate the name N and observe that this α form defines the aggregate by intensionally specifying its elements, and does not violate the "hierarchy axiom" cited above. Hence, it is a set. (Before we know that a particular aggregate is a set, we say that x is not necessarily a member of the aggregate but only an affiliate of it.) Of course, one may suppose that that axiom appears to imply infinity. But we may apply the α modifier to that axiom, without requiring an understanding of completion or closure. That is, the α modifier may be read to imply closure, but it need not be read that way.
In any case, these observations, along with Rosser's unique approach to logic in his Logic for Mathematicians (1953), reminds one that there are many roads in formal logic, not one – though of course the basics are, for the most part, unchanging.
In addition, I suppose it might be mentioned that the status of Tractatus has suffered considerable decline among mathematical logicians. Kurt Gödel noted that he had given it only "superficial" attention, though he was involved in the Vienna Circle as a young man. He minimized W's influence on that philosophical group.
By 1983, an important anthology of mathematical philosophy had dropped Tractatus from its second edition in favor of the work of others.* It seems, then, that despite Russell's sponsorship, Tractatus was more influential among those logical positivists who were not primarily mathematicians or mathematical logicians. An important exception was Rudolf Carnap,†† whose notion that logical syntax could replace outworn philosophy echoes the thrust of Tractatus. In fact, Carnap focused the Vienna Circle on Tractatus in 1926 and 1927, according to Hao Wang.‡‡ Gödel, in fact, suppressed a detailed review he wrote of Carnap's philosophy for the Library of Living Philosophers. Carnap's attempt to carry out what he took to be W's vision is not generally accepted as having succeeded.
In any case, I don't wish to undervalue Tractatus, which was thought out by a young man in a prisoner of war camp. As I mentioned, the aim here is to throw a spotlight on one of the supposed philosophical pillars of the modern educated elite, most of whom of course have only the vaguest inkling of its import.
So then, how firm is the foundation on which they stand?
Further
There is no claim here that I have fairly addressed all the abstruse points raised by W in Tractatus or that I am competent to examine the numerous contributions made by mathematical logicians since its publication. Rather, I have tried to mine this work for what nuggets there are that may be of value a century later. That is, this critique is a highly ego-centric one, I must concede.A positive summary
of Tractatus and W's other work is found at The School of Life, a YouTube channel:
https://www.youtube.com/watch?v=pQ33gAyhg2c
Freeman Dyson,
the physicist, relates a less-than-heartwarming encounter with Wittgenstein.
https://www.youtube.com/watch?v=byi3vOnVodQ
Dyson calls W a "charlatan" who liked to torment people.
Argues that W slipped up on Goedel theorem
http://wab.uib.no/agora/tools/alws/collection-6-issue-1-article-6.annotate
Kurt Gödel thought
W's later philosophy had "taken a step backward" from Tractatus, according to Hao Wang in Reflections on Kurt Gödel (MIT 1987). Wang adds that in 1972 Gödel noted that he had read Tractatus in 1927, though "never thoroughly."
†. In probability theory, the truth value system ought remain binary. Middle values pertain to degree of uncertainty. There is a case for a third truth value of 0, for provably undecidable. But, a problem with barring the logical rule of excluding the middle is that it eviscerates whole sections of mathematics that mathematicians are loath to give up.
Footnotes
++. The subset axiom is the axiom of Zermelo-Fraenkel set theory which asserts the existence for any set a and a formula A(y) of a set x consisting of all elements of a satisfying A(y).
∃x∀y (y ∈ x --> y ∈ a ● A(y))Wolfram MathWorld notes that that axiom is called the subset axiom by Enderton (1977), while Kunen (1980) calls it the comprehension axiom. Itô (1986) terms it the axiom of separation, but this name appears to not be used widely in the literature and to have the additional drawback that it is potentially confusing with the separation axioms of Hausdorff arising in topology.
This axiom was introduced by Zermelo.
Points of interest:
(i) The formula A(y) would in most cases be intensional, that is, would specify properties common to the y's.That is, "A(y)" is not itself a formula, but stands for the set of applicable formulas. Thence yo ∈ x is arbitrary. To say that an arbitrary y can be specified is to effectively express the axiom of choice. Yet, I can imagine that many would disagree with that assessment.
(ii) The axiom of choice is implied by the subset axiom.
‡ . Current physics denies that this can always hold because of the claim that some parts of the cosmos are too far away and moving too fast relative to us to ever be able to interact with our sector in any manner.
* . Philosophy of Mathematics: Selected Readings edited by Paul Benacerraf and Hilary Putnam (2d ed. Cambridge 1983; 1st ed. Prentice-Hall 1964).
† † . Reflections on Kurt Gödel by Hao Wang (MIT 1987).
‡‡ . According to André Carus's article on Carnap at the Stanford Encyclopedia of Philosophy (2020):
The Wittgensteinian program favored by the Vienna Circle ... had collapsed in 1930. But Carnap soon recovered, and during a sleepless night on 21 January 1931, conceived of an entirely new basis for the Vienna Circle’s characteristic doctrines (Awodey & Carus 2009). Instead of trying to fuse Hilbert and Wittgenstein, Carnap now dropped Wittgenstein altogether and pursued a Hilbertian approach. “Meaning” was no longer rooted in the correspondence between configurations of elementary facts and their linguistic representations. In fact, meaning was banished altogether, at least in our statements about the language of science (our metalinguistic “elucidations” such as those in the Tractatus itself or the Aufbau).
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