Could you, please, cite the line in the Tractatus that you mentioned. Thanks.
Wittgenstein's philosophy can be divided into 2/3 phases. The semantic part of the quoted phrase can be anticipated in any of the three phases. The meaning in each phase would be different as the concept of "Mathematics" is not constant over the phases.
But was the phrase ever explicitly written in that form?
It is possible to argue that Wittgenstein meant as much.
El Blog Simbiotica está lleno de artículos que niegan tal aseveración. Saludos: Alejandro Álvarez
Joel
“We can not have nothing for sure outside mathematics” is a statement that could be from Wittgenstein, but it is probably not. I wanted that you find it for yourself. Now, when the number of people started stumbling around, I will give you my answer.
Look at the online editions that you can download:
http://people.umass.edu/phil335-klement-2/tlp/tlp-ebook.pdf
http://people.umass.edu/phil335-klement-2/tlp/tlp.pdf
Tractatus in Spanish:
http://www.ub.edu/procol/sites/default/files/Wittgenstein_Tractatus_logico_philosophicus.pdf
See also the Philosophical investigations, Part II, Ch. 10, 11 and 12.
PI:
http://gormendizer.co.za/wp-content/uploads/2010/06/Ludwig.Wittgenstein.-.Philosophical.Investigations.pdf
Okay, now, dear friend; The statement should probably be: “We cannot have anything that is certain outside mathematics” or similar. It refers to the axiomatic systems that are tautologies where all statements, I mean all well formed statements (formulas) are analytically true. They however do not say anything about the world.
However the statement that you cited: is NOT from the Tractatus, it is not from Philosophical investigations, and it is not from his Cambridge lectures 1930-1032.
Of course, you are a student, but the others are not. I do not understand all of you: why don’t you look up for the answer DIRECTLY in the Wittgenstein’s works? You can read Tractatus in one day. The statement expresses the essence of logicism. Wittgenstein was already not that sure in Investigations about the completeness of mathematics. Goedel later demonstrated that such assumption cannot be demonstrated to be true.
Rene Descartes, Discourse on the Method Part1 and Meditations Part1.
Pythagoras believed this and I think hardly any of mathematicians though differently, from the times of antiquity to our days. Rare ware mathematicians who did not believe this and number of them wrote about this in various ways. Before Frege, Russell and Whitehead and the “Logicism movement” those claims were without much justification. Those claims were therefore not significant. Logicism and then Wittgenstein, Vienna circle philosophers, logical positivists and logical empiricists up to Goedel believed strongly that mathematics was self contained axiomatic system which truths were analytical.
There are however questions on RG that do not deserve answers at all.
Descartes speaks about mathematics in his Discourse on method, Part 1. He also speaks about mathematics in his Fifth meditation (and not in the “the part one”). What he says implies of course the truth of mathematics.
Discourse on method, Part 1
(...) that •mathematics contains some very subtle devices that serve not only to satisfy those who are intrigued by mathematical problems but also to help with all practical and mechanical endeavours and to lessen men’s labours;
I especially enjoyed mathematics, because of the certainty and evidentness of its reasonings.
Descartes speaks about mathematics in his Fifth meditation
And even although I had not demonstrated this, the nature of my mind is such that I could not prevent myself from holding them to be true so long as I conceive them clearly; and I recollect that even when I was still strongly attached to the objects of sense, I counted as the most certain those truths which I conceived clearly as regards figures, numbers, and the other matters which pertain to arithmetic and geometry, and, in general, to pure and abstract mathematics.
(…)
It is certain that I no less find the idea of God, that is to say, the idea of a supremely perfect Being, in me, than that of any figure or number whatever it is; and I do not know any less clearly and distinctly that an [actual and] eternal existence pertains to this nature than I know that all that which I am able to demonstrate of some figure or number truly pertains to the nature of this figure or number, and therefore, although all that I concluded in the preceding Meditations were found to be false, the existence of God would pass with me as at least as certain as I have ever held the truths of mathematics (which concern only numbers and figures) to be.
(…)
And now that I know Him I have the means of acquiring a perfect knowledge of an infinitude of things, not only of those which relate to God Himself and other intellectual matters, but also of those which pertain to corporeal nature in so far as it is the object of pure mathematics [which have no concern with whether it exists or not].
And in meditation VI:
Nothing further now remains but to inquire whether material things exist. And certainly I at least know that these may exist in so far as they are considered as the objects of pure mathematics, since in this aspect I perceive them clearly and distinctly.
This is a matter of definition. It depends what you consider "sure", "proof", and so forth.
Strictly speaking, in deductive systems, every valid conclusion is no less true ("sure") than its premises are. The same may hold in law, but in deductive systems, premises are assumptions; in law, premises should be facts. And strictly speaking, facts are never "sure".
Mario
Every sentence can be turned on its head and every meaning can be decomposed into various intentions and extentions. The above question concerns the consistency of mathemathics and referes ultimatly to the Goedels theoremes.
Joel asked (if I understood him correctly) whether some texts exist that maintain that there is no other system of axioms (or any “other” knowledge?) then mathematics that is sure (probably meaning consistent). I answered that such assumptions have been common since the antiquity. Modern logicians maintain that if the statements, i.e. propositions are tautologies, if the system is axiomatic, it will be “sure”. But not complete. Joel asked for some citations I think. Some were given. This is all. If everyone would develop here all else what s/he knows, the discussion will dissolve.
And my discussion finishes here.
Dragan,
thank you for the explanation. I did not recognize Goedel in the initial question. Anyway, I spent some time dealing with his theorem (first incompleteness, if I remember well) very long ago, but I would not have anything substantial to say about this now.
Mario
This is just fine. Thanks.
But the second applies here and it is also interesting in principle to have a discussion, I mean a discussion is needed about misuse of Gödel theorems that are often used to justify just any argument! This is similar what often has been the case also with the Relativity theory or even with "entropy"! The social scientists adopted entropy to illustrate any concept of ambiguity or sometimes just - anything.
Stefan
He, he, your comment is not bad. In fact I thought again about this and I think now that I was mistaken. But, I think, your “solution” does not really work either. There are plenty of grammatical problems with the above sentence, as I already mentioned. But let us forget them for a while.
If mathematics is a part of logic, your argument probably brakes down.
Also, if I would not utter any sentence but just refer to the propositional contents, each intentional act and its fulfillment would be a "proof" that above sentence is false.
In addition to what you wrote, the proof must be looked for in some finite world or a world that is defined. The sentence above does not define that world, so it is probably also, for the time being, meaningless.
Finally, it is true, the sentence does not say about how mathematics is, so…. So my suggestions above are not appropriate, or even false! Probably there is nothing sure in mathematic either? In the end we fall back on “cogito” problem. And that problem has no rational solution.
SUGGEST CHECKING The Division and Methods of the Sciences, Questions Four and Five of the Commentary on the De Trinitatae of Boethius, Armand Maurer.
You can not find desired proof because your proposition is false. There exist practically infinite set of propositions for sure outside of mathematics. I give you only one, but looking at this you will find as many analogous examples as you want.
Now, here is such truth:
Eugene Onegin killed Vladimir Lensky.
P. S. Added 06.01.2016 at 12:48. In a private letter, I was asked to explain my thought. Well, the given proposition is true inside of Pushkin's novel "Eugene Onegin" and this novel is outside of mathematics.
The sentence's grammar is confusing. "We can not have nothing...: Would it be more correct to ask "Can we know anything as equally certain as our knowledge of mathematics?"
Charles Cassini
Your question is rather different. Joel Torres asked about proposition while your ask is about knowledge.
Charles
Could you please give us the full reference of The division and methods of the sciences, including pages, so we can have a look? Please.
Gramática aparte, la cuestión es muy controversial, todo es 'de acuerdo al color del cristal con que miremos'
Dear Joel,
Could you tell me, please, where inside Tractatus you found the proof of your false proposition? To the best of my knowledge, the Tractatus consists only of aphorisms and has no proofs.
Dragan says: ... a discussion is needed about misuse of Gödel theorems that are often used to justify just any argument! This is similar what often has been the case also with the Relativity theory or even with "entropy"!
This is very true; but I am very busy now with a text on which I work; I answered this question because RG offered it to me. However, I cannot discuss it further now.
Mario
No problem, I am terribly busy also.
Aslanbek
Joel asked about the books that probably contain such affirmation, that’s all.
There are plenty of texts and books, as I indicated above, that in a way confirm that outside mathematics there is not very much that could be taken seriously as “sure”.
The Tractatus certainly contains number of proofs. You may look up on line; I gave above in one of my answers number of links to the Tractatus, even in German. Please read Tractatus, it is short and could be read in a day.
Above, I explained what a probably correctly formulated proposition of Joel implies. It states that the propositions that are synthetic (conjectures about the matters of facts in the world) are not sure. They may be true but it is not certain since we never can satisfy a demand of correspondence of the proposition with how the world is. In axiomatic systems, in deductive logic and in great part of mathematics, the truths are analytical; there is nothing in a conclusion that is not already in the premises.
Here I give it again:
Look at the online editions that you can download:
http://people.umass.edu/phil335-klement-2/tlp/tlp-ebook.pdf
http://people.umass.edu/phil335-klement-2/tlp/tlp.pdf
Tractatus in Spanish:
http://www.ub.edu/procol/sites/default/files/Wittgenstein_Tractatus_logico_philosophicus.pdf
See also the Philosophical investigations, Part II, Ch. 10, 11 and 12.
PI:
http://gormendizer.co.za/wp-content/uploads/2010/06/Ludwig.Wittgenstein.-.Philosophical.Investigations.pdf
The statement should probably be: “We cannot have anything that is certain outside mathematics” or similar. It refers to the axiomatic systems that are tautologies where all statements, I mean all well formed statements (formulas) are analytically true. They however do not say anything about the world.
However the statement that you cited: is NOT from the Tractatus, it is not from Philosophical investigations, and it is not from his Cambridge lectures 1930-1932.
OK, but, maybe, you tell us what for was all previous discussion and what did you learn about your proposition: is it true or false?
Joel
Try to find a companion to Wittgenstein Tractatus or:
https://books.google.ca/books/about/A_Companion_to_Wittgenstein_s_Tractatus.html?id=uzY9AAAAIAAJ
Or SparkNotes may do as well:
http://www.sparknotes.com/philosophy/tractatus/
Best. wishes.
D.
Aslanbek
This is where my recent comment was. I removed it because you were so kind to remove yours. Thanks very much indeed.
Dear Dragan,
Excuse me, please, for a delay with answer --- I was away from the early morning.
I'm sorry that you saw something offensive in my previous comment. Please be sure that I did not want something like. Expenses of temperament, not more. So I without regrets deleted this comment.
After that, maybe you will permit me to say a couple of words about one of your last remarks.
Wittgenstein did not call 6.241 "a proof". 6.241, as well as all in Tractatus, is aphorism. It IS NOT a proof, it only MENTIONED a proof. And, of course, this (mentioned) proof is not what Joel was looking for.
Thank you for your useful, fruitful and instructive advice.
Glad to meet you.
Sincerely yours,
Aslanbek
No problem, Aslanbek.
(I obtained one useless degree in Logic that I never mention in my CV. This is why I do not pay much attention to what other people claim to have as "degrees". I look at the arguments and their acheivements, articles.... I recommend to you to do the same. Labels do not help solve the problems. Knowledge does. Wittgenstein never studied philosophy also.)
.
The 6.241 is a proof (probably contains a mistake, but is a proof). There is an explanation, that is not easy, in:
Wittgenstein's Philosophy of Mathematics
By Pasquale Frascolla, page 13 and later. I recomend reading all chapter.
Stefan
On the contrary, we agree.
Take that my intentional state is: a finite set of well-formed and formal-logically consistent axioms.
And that my fulfillment of the above is: a mechanized or at least in principle mechanizable finite sequence of logical rule-based derivation steps.
Then - we would then fully agree.
Unfortunately I am not very much “into” the Proof Theory and the subject is too large. Therefore I would not really go into it in the near future. It is possible that Aslanbek understands more about this so you can probably discuss this with him.
Dragan
1. About your recommendation. I always try to do the same. But this time... Your words "Please read Tractatus..." brought me out of balance, hence my reaction.
2. About Wittgenstein's 6.241. I can't understand why do you think that 6.241 IS a proof. In the aphorism 6.24 Wittgenstein makes the general statement:
The method by which mathematics arrives at its equations is the method of substitution.
And 6.241 is subordinated to 6.24 aphorism in which W. clarifies 6.24 by an example of proof from arithmetics. Thus, 6.241 is NOT a proof of 6.24 at least for the general statement cannot be proved by an example.
One would say that 6.241 CONTAINS the proof (of the proposition "2x2 = 4") but this is not true as well. Speaking about the proof W. had to enclose this proof in quotes. So, the correct formulation of 6.241 does NOT contain the proof, it contains the QUOTATION of the proof. Now we just have to remember the remarkable Quine's example: "Boston" does no contain Boston for Boston contains some 800,000 people while "Boston" contains 6 letters and does not contain people.
Those who prefer not to use quotes must distinguish between use and mention. Anyway, we see that 6.241 IS NOT a proof and DOES NOT CONTAIN a proof, it only MENTIONS a proof as I already said in one of my previous comments.
Aslanbek
OK. But I think that Wittgenstein did not make that distinction and was presenting a proof, similarly as Russell was doing in the "Principia". The later analyses of "Tractatus", all that I either have or have seen, treat this or other statements from the “Tractatus” in the same way. Russell himself, in the introduction, recommends that we take the text as clarifying the relations between sentences as “symbols” and I think we should take them as such, without any other assumptions.
Best wishes.
Dragan
"Wittgenstein was presenting a proof "
A proof of what? --- Of a statement from arithmetics, not of his own statement.
"Russell himself, in the introduction, recommends that we take the text as clarifying "
Just so! As clarifying, not as proving! Precisely this I have in mind when I say that Tractatus does not contain proofs. It contains no proofs of Wittgenstein's statements, only clarifications.
Dear Joel!
You can find it in "on what there is" which is famous ontological article of Quine.
He argues about Existence of things, such as numbers and mathematical entities.
https://tu-dresden.de/die_tu_dresden/fakultaeten/philosophische_fakultaet/iph/thph/braeuer/lehre/metameta/Quine%20-%20On%20What%20There%20Is.pdf
Steffan
Nice comments, but I do not agree.
Wittgenstein maintains that logic and mathematics are NOT about the external world; he maintains that there are proofs of logic that of course do not prove anything outside logic. The discussion here was about meta-language and the "proof" INSIDE Tractatus, and there are (this is my opinion) "proofs" in the Tractatus of various kinds (deductive, sequential, axiomatic, etc.) (I already said that the "Proof theory" is not my expertise at all, but this is just suficiently general, so I have "an opinion"); tautology is for Wittgenstein the ultimate proof that does not need explanation. Not just modus ponens but a simple proposition is already proof for him. There are plenty in Tractatus.
Sorry, I do not have time now to develop further. Your comment was really excellent, and therefore needs really good answer. But you put so much there that while an answer must be long - yet in general I simply do not think that this corresponds to the reality. You write that he hoped for:
“1-to-1 bijective correspondence between linguistic atoms (propositions) and atomic 'facts' in the extra-linguistic external world.”
Why do you say this? He explicitly denies this. He certainly does not think that the correspondence of mental concepts and the reality is their “identity”. Also, I do not know Nietzsche’s work so well. But what you say about Heidegger is simply not true.
I admit it is unfair not to advance arguments. Sorry, really. May be I will do this later this month.
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