Wittgenstein Vs Turing: Logic Contradictions

Posted by Ali Reda | Posted in | Posted on 10/26/2013

Turing attended Wittgenstein's lectures on the philosophy of mathematics in Cambridge in 1939 and disagreed strongly with a line of argument that Wittgenstein was pursuing which wanted to allow contradictions to exist in mathematical systems. Wittgenstein argues that he can see why people don't like contradictions outside of mathematics but cannot see what harm they do inside mathematics.

Wittgenstein: Think of the case of the Liar: It is very queer in a way that this should have puzzled anyone — much more extraordinary than you might think... Because the thing works like this: if a man says 'I am lying' we say that it follows that he is not lying, from which it follows that he is lying and so on. Well, so what? You can go on like that until you are black in the face. Why not? It doesn't matter. ...it is just a useless language-game, and why should anyone be excited? 
Turing: What puzzles one is that one usually uses a contradiction as a criterion for having done something wrong. But in this case one cannot find anything done wrong. 
Wittgenstein: Yes — and more: nothing has been done wrong, ... where will the harm come? 
Turing: The real harm will not come in unless there is an application, in which a bridge may fall down or something of that sort…. You cannot be confident about applying your calculus until you know that there are no hidden contradictions in it. 
Wittgenstein: There seems to me an enormous mistake there. ... Suppose I convince [someone] of the paradox of the Liar, and he says, 'I lie, therefore I do not lie, therefore I lie and I do not lie, therefore we have a contradiction, therefore 2x2 = 369.' Well, we should not call this 'multiplication,' that is all... 
Turing: Although you do not know that the bridge will fall if there are no contradictions, yet it is almost certain that if there are contradictions it will go wrong somewhere. 
Wittgenstein: But nothing has ever gone wrong that way yet...

Turing is exasperated and points out that such contradictions inside mathematics will lead to disasters outside mathematics: bridges will fall down. Only if there are no applications will the consequences of contradictions be innocuous. Turing eventually gave up attending these lectures. His despair is understandable. The inclusion of just one contradiction (like 0 = 1) in an axiomatic system allows any statement about the objects in the system to be proved true (and also proved false).

When Bertrand Russel pointed this out in a lecture he was once challenged by a heckler demanding that he show how the questioner could be proved to be the Pope if 2 + 2 = 5. Russel replied immediately that 'if twice 2 is 5, then 4 is 5, subtract 3; then 1 = 2. But you and the Pope are 2; therefore you and the Pope are 1'! A contradictory statement is the ultimate Trojan horse.

Therefore classical logic is not always applicable to real-world situations, no matter how well the parts we've used so far seem to have worked. And we have no theory of when it will be applicable and when it will fail (at least, we didn't in Wittgenstein's time; some might argue that relevance logics give us such a theory now).


Posted by Ali Reda | Posted in | Posted on 10/25/2013

"Dasein," the being who understands Being. Existence means that Dasein is potentiality-for-being; it projects its being upon various possibilities. Existence represents thus the phenomenon of the future. Then, as thrownness, Dasein always finds itself already in a certain spiritual and material, historically conditioned environment; in short, in the world, in which the space of possibilities is always somehow limited. This represents the phenomenon of the past as having-been. Finally, as fallenness, Dasein in the world exists in the midst of beings which are both Dasein and not Dasein. This represents the primordial phenomenon of the present. Accordingly, Dasein is not temporal for the mere reason that it exists “in time,” but because its very being is rooted in temporality: the original unity of the future, the past and the present which constitutes authentic temporality.

According to Heidegger, as this sense of being precedes any notions of how or in what manner any particular being or beings exist, it is pre-conceptual, non-propositional, and hence pre-scientific.Thus, in Heidegger's view, fundamental ontology would be an explanation of the understanding preceding any other way of knowing, such as the use of logic, theory, specific ontology or act of reflective thought. Being is to be grasped by means of the phenomenological method. One must direct oneself toward a Dasein  but in such a way that its being is thereby brought out.

Metaphysics provides an answer to the question of the being of beings for contemporary men and women, but skillfully removes from their lives the problem of their own existence. Metaphysics cannot be rejected, canceled or denied, but it can be overcome by demonstrating its forgetfulness of being.

Heidegger talks about "enframing". You place a frame around something and it brings qualities of understanding, revealing aspects of our humanity or aspects of our universe. At one time, when we worked with our hands, technology was a means of enframing through which we discovered things about ourselves. The early making of technological devices to explore scientific reality was a driving force in scientific research.

But enframing can work the other way as well, concealing things from our attention. For example, a friendly chat in the bar is turned into networking. Heidegger claims that what is “horrifying” is not any of technology’s particular harmful effects but “what transposes ... all that is out of its previous essence” — that is to say, what is dangerous is that technology displaces beings from what they originally were, hindering our ability to experience them truly. Modern technology has separated us from direct experience of the world. Heidegger’s alternatives provide ways to clarify the irreducibility of our experience to what we can capture technologically, or through natural science.

Heidegger observes that because of technology, “all distances in time and space are shrinking” and “yet the hasty setting aside of all distances brings no nearness; for nearness does not consist in a small amount of distance.” In order to experience nearness, we must encounter things in their truth. And no matter how much we believe that science will let us “encounter the actual in its actuality,” science only offers us representations of things. It “only ever encounters that which its manner of representation has previously admitted as a possible object for itself.”

Scientifically speaking, the distance between a house and the tree in front of it can be measured neutrally: it is thirty feet. But in our everyday lives, that distance is not as neutral, not as abstract. Instead, the distance is an aspect of our concern with the tree and the house: the experience of walking, of seeing the tree’s shape grow larger as I come closer, and of the growing separation from the home as I walk away from it. In the scientific account, “distance appears to be first achieved in an opposition” between viewer and object. By becoming indifferent to things as they concern us, by representing both the distance and the object as simple but useful mathematical entities or philosophical ideas, we lose our truest experience of nearness and distance.

Wittgenstein: A Life

Posted by Ali Reda | Posted in | Posted on 10/24/2013

“You can't think decently if you're not willing to hurt yourself”


Wittgenstein was born in 1889 into one of the wealthiest families in the Austro-Hungarian empire. His father was a major figure in the Austrian iron and steel industry, Ludwig grew up in an environment in which the intellectual and artistic currents of the cultural life of Vienna were dominant. Ludwig, who was the youngest of the family. The children were baptized as Catholics, and raised in an exceptionally intense environment. The family was at the center of Vienna's cultural life; Bruno Walter described the life at the Wittgensteins' palace as an "all-pervading atmosphere of humanity and culture". Music was a part of the daily life of the home.

Psychiatrist Michael Fitzgerald argues that Karl (Father) was a harsh perfectionist who lacked empathy, and that Wittgenstein's mother was anxious and insecure, unable to stand up to her husband. Three of the five brothers would later commit suicide.

It was while he was at the Realschule that he decided he had lost his faith in God. He discussed it with Gretl, his other sister, who directed him to Arthur Schopenhauer's The World as Will and Representation. As a teenager, Wittgenstein adopted Schopenhauer's epistemological idealism. However, after his study of the philosophy of mathematics, he abandoned epistemological idealism for Gottlob Frege's conceptual realism. In later years, Wittgenstein was highly dismissive of Schopenhauer, describing him as an ultimately "shallow" thinker: "Schopenhauer has quite a crude mind... where real depth starts, his comes to an end"

While a student at the Realschule, Wittgenstein was influenced by Austrian philosopher Otto Weininger's 1903 book Geschlecht und Charakter (Sex and Character). Weininger (1880–1903), who was both Jewish and homosexual, argued that the concepts male and female exist only as Platonic forms, and that Jews tend to embody the platonic femininity. Whereas men are basically rational, women operate only at the level of their emotions and sexual organs. Jews, Weininger argued, are similar, saturated with femininity, with no sense of right and wrong, and no soul. The only life worth living is the spiritual one—to live as a woman or a Jew means one has no right to live at all; the choice is genius or death. Weininger committed suicide, shooting himself in 1903, shortly after publishing the book. Many years later, as a professor at Cambridge, Wittgenstein distributed copies of Weininger's book to his bemused academic colleagues. He said that Weininger's arguments were wrong, but that it was the way in which they were wrong that was interesting.

Ludwig at first prepared himself for a career in engineering. He enrolled in and did research at the Engineering Laboratory of the University of Manchester, where he remained until the winter term of 1911. Wittgenstein came upon Bertrand Russell's Principles of Mathematics, published in 1903. He read this with great interest. Through the study of this book he learned of Frege's `new logic'. Wittgenstein became so absorbed in these studies that he decided to give up aeronautical engineering as a career. In the summer of 1911 he visited Frege at the University of Jena to show him some philosophy of mathematics and logic he had written, and to ask whether it was worth pursuing. He wrote:
I was shown into Frege's study. Frege was a small, neat man with a pointed beard who bounced around the room as he talked. He absolutely wiped the floor with me, and I felt very depressed; but at the end he said 'You must come again', so I cheered up.
On Frege's suggestion he went to Cambridge to study with Russell, so on 18 October 1911 Wittgenstein arrived unannounced at Russell's rooms in Trinity College. Russell was having tea with C. K. Ogden, he was soon not only attending Russell's lectures, but dominating them. The lectures were poorly attended. Wittgenstein started following him after lectures back to his rooms to discuss more philosophy, until it was time for the evening meal in Hall. Russell grew irritated; he wrote to his lover Lady Ottoline Morrell:
"My German friend threatens to be an infliction". 
Russell wrote in November 1911 that he had at first thought Wittgenstein might be a crank, but soon decided he was a genius:
 "Some of his early views made the decision difficult. He maintained, for example, at one time that all existential propositions are meaningless. This was in a lecture room, and I invited him to consider the proposition: 'There is no hippopotamus in this room at present.' When he refused to believe this, I looked under all the desks without finding one; but he remained unconvinced." 
Three months after Wittgenstein's arrival Russell told Morrell:
"I love him & feel he will solve the problems I am too old to solve ... He is the young man one hopes for."
 Russell, speaking to Wittgenstein's sister, Hermine, 1912:
We expect the next big step in philosophy to be taken by your brother.
And in his Autobiography, He wrote:
He was perhaps the most perfect example I have ever known of genius as traditionally conceived, passionate, profound, intense, and dominating. He had a kind of purity which I have never known equalled except by G. E. Moore. I remember taking him once to a meeting of the Aristotelian Society, at which there were various fools whom I treated politely. When we came away he raged and stormed against my moral degradation in not telling these men what fools they were.
At the end of his first term at Trinity, he came to me and said: `Do you think I am an absolute idiot?' I said: `Why do you want to know?' He replied: 'Because if I am I shall become an aeronaut, but if I am not I shall become a philosopher.' I said to him: `My dear fellow, I don't know whether you are an absolute idiot or not, but if you will write me an essay during the vacation upon any philosophical topic that interests you, I will read it and tell you.' He did so, and brought it to me at the beginning of the next term. As soon as I read the first sentence, I became persuaded that he was a man of genius, and assured him that he should on no account become an aeronaut.
He used to come to see me every evening at midnight, and pace up and down the room like a wild beast for three hours in agitated silence. Once I said to him: 'Are you thinking about logic, or about your sins?' 'Both', he replied, and continued his pacing. I did not like to suggest it was time for bed, for it seemed probable both to him and to me that on leaving me he would commit suicide.
 The role-reversal between him and Wittgenstein was such that he wrote in 1916, after Wittgenstein had criticized his own work:
"His criticism, 'tho I don't think he realized it at the time, was an event of first-rate importance in my life, and affected everything I have done since. I saw that he was right, and I saw that I could not hope ever again to do fundamental work in philosophy." 
Wittgenstein remained for the three terms of 1912 and for the first two terms of 1913. Pinsent wrote in May 1912 that Wittgenstein had just begun to study the history of philosophy:
"He expresses the most naive surprise that all the philosophers he once worshipped in ignorance are after all stupid and dishonest and make disgusting mistakes!"
Karl Popper, another Viennese philosopher, had been invited as the guest speaker. Popper's paper was "Are there philosophical problems?", in which he struck up a position against Wittgenstein's, contending that problems in philosophy are real, not just linguistic puzzles as Wittgenstein argued. Accounts vary as to what happened next, but Wittgenstein apparently started waving a hot poker, demanding that Popper give him an example of a moral rule. Popper offered one—"Not to threaten visiting speakers with pokers"—at which point Russell told Wittgenstein he had misunderstood and Wittgenstein left.

In the autumn of 1913 Wittgenstein went to Norway with a young mathematician friend from Cambridge, David Pinsent. Pinsent's diaries provide valuable insights into Wittgenstein's personality - sensitive, nervous and attuned to the tiniest slight or change in mood from Pinsent. Pinsent was later killed in World War I, and the Tractatus is dedicated to him. In a letter to Russell in 1913:
“My day passes between logic, whistling, going for walks, and being depressed. I wish to God that I were more intelligent and everything would finally become clear to me - or else that I needn’t live much longer.”
deep inside me there's a perpetual seething, like the bottom of a geyser, and I keep hoping that things will come to an eruption once and for all, so that I can turn into a different person.
Perhaps you regard this thinking about myself as a waste of time - but how can I be a logician before I'm a human being? Far the most important thing is to settle accounts with myself!
He later saw this as one of the most productive periods of his life, writing Logik (Notes on Logic), the predecessor of much of the Tractatus. He used to say:
 'Then my mind was on fire!'
While in Norway, Wittgenstein learned Norwegian to converse with the local villagers, and Danish to read the works of the Danish philosopher Søren Kierkegaard. He remained there until the outbreak of the war in 1914.

Moore asked the university to consider accepting Logik as sufficient for a bachelor's degree, but they refused, saying it wasn't formatted properly: no footnotes, no preface. Wittgenstein was furious, writing to Moore in May 1914:
"If I am not worth your making an exception for me even in some STUPID details then I may as well go to Hell directly; and if I am worth it and you don't do it then—by God—you might go there."
Wittgenstein volunteered for military service in the Austrian army and saw active service at the front, yet managed to find time during this period to do some writing. In March 1916, he was posted to a fighting unit on the front line of the Russian front, as part of the Austrian 7th Army, where his unit was involved in some of the heaviest fighting, defending against the Brusilov Offensive. In action against British troops, he was decorated with the Military Merit with Swords on the Ribbon, and was commended by the army for "His exceptionally courageous behaviour, calmness, sang-froid, and heroism", which "won the total admiration of the troops." In January 1917, he was sent as a member of a howitzer regiment to the Russian front, where he won several more medals for bravery including the Silver Medal for Valour, First Class. In 1918, he was promoted to lieutenant and sent to the Italian front as part of an artillery regiment.

In 1916 Wittgenstein read Dostoevsky's The Brothers Karamazov so often that he knew whole passages of it by heart, particularly the speeches of the elder Zossima, who represented for him a powerful Christian ideal, a holy man "who could see directly into the souls of other people".  Russell said he returned from the war a changed man, one with a deeply mystical and ascetic attitude. A series of events around this time left him deeply upset. On 13 August, his uncle Paul died. On 25 October, he learned that Jahoda and Siegel had decided not to publish the Tractatus, and on 27 October, his brother Kurt killed himself, the third of his brothers to commit suicide. It was around this time he received a letter from David Pinsent's mother to say that Pinsent had been killed in a plane crash on 8 May.

He carried the manuscript of his work in his rucksack, and it was with him when he was captured at the end of the war, was made a prisoner, and confined at Monte Cassino in November 1918. He subsequently spent nine months in an Italian prisoner of war camp. He returned to his family in Vienna on 25 August 1919, by all accounts physically and mentally spent. He apparently talked incessantly about suicide, terrifying his sisters and brother Paul. In a letter to Russell:
The best for me, perhaps, would be if I could lie down one evening and not wake up again.
He decided to do two things: to enroll in teacher training college as an elementary school teacher, and to get rid of his fortune.

Dear Russell,
[Cassino, Provincia Caserta, Italy]
Thanks so much for your postcards dated 2°' and 3rd of March. I've had a very bad time, not knowing whether you were dead or alive! I can't write on Logic as I'm not allowed to write more than two post] c[ard]s a week (15 lines each). This letter is an exception, it's posted by an Austrian medical student who goes home tomorrow. I've written a book called "Logisch-Philosophische Abhandlung" containing my work of the last six years. I believe I've solved our problems finally. This may sound arrogant but I can't help believing it. I finished the book in August 1918 and two months after was made Prigioniere. I've got the manuscript here with me. I wish I could copy it out for you; but its pretty long and I would have no safe way of sending it to you. In fact you would not understand it without a previous explanation as it's written in quite short remarks. (this of course means that nobody will understand it; although I believe, it's all as clear as crystal. But it upsets all our theory of truth, of classes, of numbers and all the rest.) I will publish it as soon as I get home. Now I'm afraid this won't be "before long". And consequently it will be a long time yet till we can meet. I can hardly imagine seeing you again!
It will be too much! I suppose it would be impossible for you to come and see me here? or perhaps you think it's colossal cheek of me even to think of such a thing. But if you were on the other end of the world and I could come to you I would do it.
Please write to me how you are, remember me to Dr. Whitehead. Is old Johnson still alive? Think of me often!
Ever yours
Ludwig Wittgenstein 

I got a letter from him written from Monte Cassino, saying that a few days after the Armistice, he had been taken prisoner by the Italians, but fortunately with his manuscript. It appears he had written a book in the trenches, and wished me to read it. He was the kind of man who would never have noticed such small matters as bursting shells when he was thinking about logic. ... It was the book which was subsequently published under the title Tractatus Logico-Philosophicus. ~ Bertrand Russell in The Autobiography of Bertrand Russell (1968) Ch. 9 : Russia, p. 330
The story of his repeated frustrations in trying to get his manuscript published at the end of the war marks one among the many unhappy chapters in his life. Wittgenstein desperately and repeatedly sought a publisher for his book and was turned down by five publishers! It narrowly missed not being published at all. Russell had agreed to write an introduction to explain why it was important, because it was otherwise unlikely to have been published: it was difficult if not impossible to understand, and Wittgenstein was unknown in philosophy. It finally appeared in 1922 in English. But it was not understood at all by Frege, and fundamentally misunderstood by Russell. In a letter written to Russell from the prison camp to which Wittgenstein was confined at the end of the World War I, there occurs the following passage:
Now I'm afraid you haven't really got hold of my main contention, to which the whole business of logical propositions is only a corollary. The main point is the theory of what can be expressed by propositionls i.e. by language (and, which comes to the same, what can be thought) and what can not be expressed by propositions, but only shown; which, I believe, is the cardinal problem of philosophy. I also sent my M.S. to Frege. He wrote me a week ago and I gather that he doesn't understand a word of it all. So my only hope is to see you soon and explain all to you, for it is VERY hard not to be understood by a single soul!
After his military service during the First World War, he decided to become an elementary school teacher. He attended teacher training college in the Kundmanngasse in Vienna in September 1919, and in 1920 was given his first job as a teacher in Trattenbach, a village of just a few hundred inhabitants about 90 km southwest of Vienna in Lower Austria. It appears that he did not have a high opinion of the villagers. Writing to Bertrand Russell, Wittgenstein said:
"I am still at exile, surrounded, as ever, by odiousness and baseness. I know that human beings on the average are not worth much anywhere, but here they are much more good-for-nothing and irresponsible than elsewhere". 
A year later, He wrote to Russell again saying:
"I am now in another hole, though I have to say, it is no better than the old one. Living with human beings is hard!"
There was one boy in particular, Karl Gruber, from an impoverished family with six children, whom Wittgenstein wanted to adopt. The two studied Latin, Greek, and mathematics together from four to seven in the evening, then dined in Wittgenstein's room. He proposed the adoption arrangement to Karl's parents, offering to send the boy to the city and finance his education. The boys' mother agreed, but the father said no and called Wittgenstein "ein verrückter Kerl" ("a crazy fellow").

Wittgenstein was reportedly seen as a tyrant by the slower students, boxing ears and pulling hair. Wittgenstein was a very nervous teacher. He would break out in a sweat, rub his chin a lot, pull his hair, and bite into a crumpled handkerchief. A student from Cambridge, Frank P. Ramsey, arrived in Austria to visit him on 17 September 1923 to discuss a review of the Tractatus he had agreed to write for Mind. Ramsey told John Maynard Keynes back in Cambridge that Wittgenstein was refusing all financial help from his family, and was even returning Christmas presents they sent him, because he did not want to have any money he had not earned himself.

Josef Haidbauer was one of Wittgenstein's pupils, 11 years old and by all accounts a weak child and a slow learner. His father had died, and his mother worked locally as a maid for a farmer named Piribauer. Piribauer himself had a daughter, Hermine, in Wittgenstein's class; Wittgenstein had reportedly pulled her so hard by the ears and hair that her ears had bled, and some of her hair had fallen out. During a lesson in April 1926 Wittgenstein hit Haidbauer two or three times on the head, and the boy collapsed unconscious. Wittgenstein sent the class home, carried Haidbauer to the headmaster's office, then quickly left the building. He bumped into Herr Piribauer on the way out, who had arrived at the school after the children alerted him. Piribauer said that when he met Wittgenstein in the hall that day: "I called him all the names under the sun. I told him he wasn't a teacher, he was an animal trainer! And that I was going to fetch the police right away!". Piribauer tried to have Wittgenstein arrested, but the one-man police station was empty when he went there, and the next day he was told Wittgenstein had disappeared. On 28 April, Wittgenstein handed in his resignation to Wilhelm Kundt, a local school inspector. He returned to Vienna, where he took a job as an assistant gardener in the Brothers of Mercy monastery in Hütteldorf. Wittgenstein was summoned to appear before the district court in Gloggnitz on 17 May 1926. Waugh writes that he lied to the court about his use of physical punishment against the children. The judge suspected he was mentally ill, and ordered an adjournment for psychiatric reports. Wittgenstein's family was one of the wealthiest in Europe at the time, and Waugh writes that they may have managed to cover things up.

The Tractatus was now the subject of much debate amongst philosophers, and Wittgenstein was a figure of increasing international fame. In particular, a discussion group of philosophers, scientists and mathematicians, known as the Vienna Circle, had built up largely as a result of the inspiration they had been given by reading the Tractatus. From 1926, with the members of the Vienna Circle, Wittgenstein would take part in many discussions. However, during these discussions, it soon became evident that Wittgenstein held a different attitude towards philosophy than the members of the Circle whom his work had inspired. For example, during meetings of the Vienna Circle, he would express his disagreement with the group's misreading of his work by turning his back to them and reading poetry aloud.

Rudolf Carnap said about him at that time:
"His point of view and his attitude toward people and problems, even theoretical problems, were much more similar to those of a creative artist than to those of a scientist; one might almost say, similar to those of a religious prophet or a seer. When he started to formulate his view on some specific problem, we often felt the internal struggle that occurred in him at that very moment, a struggle by which he tried to penetrate from darkness to light under an intense and painful strain, which was even visible on his most expressive face. When finally, sometimes after a prolonged arduous effort, his answers came forth, his statement stood before us like a newly created piece of art or a divine revelation"
Ten years later, Wittgenstein was living in Norway, and went through a period of wanting to make confessions to his friends about various issues, one of which was his use of violence against the children in Austria. One of the friends he confessed to, Fania Pascal, recalled the confession as: 
"During the short period when he was teaching at a village school in Austria, he hit a little girl in his class and hurt her (my memory is, without details, of a physically violent act). When she ran to the headmaster to complain, Wittgenstein denied that he had done it." 
In the same year that he made this confession to friends, he also travelled to Otterthal and appeared without warning at the homes of the children he had hurt. He visited at least four of them, asking for their forgiveness. In 1937 he wrote in a notebook:
Last year with God's help I pulled myself together and made a confession. This brought me into more settled waters, into a better relation with people, and to a greater seriousness. But now it is as though I had spent all that, and I am not far from where I was before. I am cowardly beyond measure. If I do not correct this, I shall again drift entirely into those waters through which I was moving then.
In 1926, Wittgenstein was again working as a gardener for a number of months, this time at the monastery of Hütteldorf, where he had also enquired about becoming a monk. His sister, Margaret, invited him to help with the design of her new townhouse in Vienna's Kundmanngasse. Wittgenstein, his friend Paul Engelmann, and a team of architects developed a spare modernist house. In particular, Wittgenstein focused on the windows, doors, and radiators, demanding that every detail be exactly as he specified. When the house was nearly finished Wittgenstein had an entire ceiling raised 30mm so that the room had the exact proportions he wanted.

At the urging of Ramsey and others, Wittgenstein returned to Cambridge in 1929. Keynes wrote in a letter to his wife:
"Well, God has arrived. I met him on the 5.15 train." 
Despite this fame, he could not initially work at Cambridge as he did not have a degree, so he applied as an advanced undergraduate. Russell noted that his previous residency was sufficient for a PhD, and urged him to offer the Tractatus as his thesis. It was examined in 1929 by Russell and Moore; at the end of the thesis defence, Wittgenstein clapped the two examiners on the shoulder and said,
"Don't worry, I know you'll never understand it." 
Moore wrote in the examiner's report:
"I myself consider that this is a work of genius; but, even if I am completely mistaken and it is nothing of the sort, it is well above the standard required for the Ph.D. degree."
Wittgenstein was appointed as a lecturer and was made a fellow of Trinity College.

In the Cambridge years, the philosopher Karl Britton said:
Wittgenstein spoke without notes but knew very well what he wanted to discuss and what he wanted to "put across," though sometimes he seemed to change his mind on some point while he was speaking . . . . But the most characteristic of all his attitudes was a very quiet, very intense stare-suddenly adopted and leading to a slow deliberate utterance of some new point. Very often he got thoroughly "stuck": appealed in vain to his hearers to help him out: he would walk about in despair murmuring: "I'm a fool, I'm a fool." And such was the difficulty of the topics he discussed, that all this struggle did not seem to us to be in the least excessive.
After the Anschluss, his brother Paul left almost immediately for England, and later the US. The Nazis discovered his relationship with Hilde Schania, a brewer's daughter with whom he had had two children but whom he had never married, though he did later. Because she was not a Jew, he was served with a summons for Rassenschande (racial defilement). He told no one he was leaving the country, except for Hilde who agreed to follow him. Wittgenstein began to investigate acquiring British or Irish citizenship with the help of Keynes, and apparently had to confess to his friends in England that he had earlier misrepresented himself to them as having just one Jewish grandparent, when in fact he had three.

From 1936 to 1937, Wittgenstein lived again in Norway,where he worked on the Philosophical Investigations. After G. E. Moore resigned the chair in philosophy in 1939, Wittgenstein was elected, and acquired British citizenship soon afterwards. He grew angry when any of his students wanted to become professional philosophers. In September 1941 he asked John Ryle, the brother of the philosopher Gilbert Ryle, if he could get a manual job at Guy's Hospital in London. John Ryle was professor of medicine at Cambridge and had been involved in helping Guy's prepare for the Blitz. Wittgenstein told Ryle he would die slowly if left at Cambridge, and he would rather die quickly. He started working at Guy's shortly afterwards as a dispensary porter, meaning that he delivered drugs from the pharmacy to the wards. He resigned the professorship at Cambridge in 1947 to concentrate on his writing, and in 1947 and 1948 travelled to Ireland where he began the manuscript volume MS 137, then he went to USA. He returned to London, where he was diagnosed with an inoperable prostate cancer, which had spread to his bone marrow.
"I won't say 'See you tomorrow' because that would be like predicting the future, and I'm pretty sure I can't do that." (1949)
He spent the next two months in Vienna, where his sister Hermine died on 11 February 1950; he went to see her every day, but she was hardly able to speak or recognize him. He wrote:
"Great loss for me and all of us, greater than I would have thought." 
On 27 November he moved into "Storey's End", at 76 Storey's Way, the home of his doctor, Edward Bevan, and his wife Joan; he had told them he did not want to die in a hospital, so they said he could spend his last days in their home instead. Joan at first was afraid of Wittgenstein, but they soon became good friends.

By the beginning of 1951, it was clear that he had little time left. He wrote a new will in Oxford on 29 January, naming Rhees as his executor, and Anscombe and von Wright his literary administrators, and wrote to Norman Malcolm that month to say:
"My mind's completely dead. This isn't a complaint, for I don't really suffer from it. I know that life must have an end once and that mental life can cease before the rest does."
In February he returned to the Bevans' home to work on MS 175 and MS 176. These and other manuscripts were later published as Remarks on Colour and On Certainty. It was his 62nd birthday on 26 April. He went for a walk the next afternoon, and wrote his last entry that day, 27 April. That evening, he became very ill; when his doctor told him he might live only a few days, he reportedly replied, "Good!" Joan stayed with him throughout that night, and just before losing consciousness for the last time on 28 April, he told her:
"Tell them I've had a wonderful life"
Philosophical Investigations was published in two parts in 1953. Most of Part I was ready for printing in 1946, but Wittgenstein withdrew the manuscript from his publisher. The shorter Part II was added by his editors, Elizabeth Anscombe and Rush Rhees.

Bertrand Russell in My Philosophical Development, Wrote of his later philosophy:
The later Wittgenstein, on the contrary, seems to have grown tired of serious thinking and to have invented a doctrine which would make such an activity unnecessary. I do not for one moment believe that the doctrine which has these lazy consequences is true. I realize, however, that I have an overpoweringly strong bias against it, for, if it is true, philosophy is, at best, a slight help to lexicographers, and at worst, an idle tea-table amusement


Tractatus Logico-Philosophicus

For a large class of cases — though not for all — in which we employ the word meaning it can be explained thus: the meaning of a word is its use in the language.
“Philosophy is a battle against the bewitchment of our intelligence by means of language.”
“A serious and good philosophical work could be written consisting entirely of jokes.”
Uttering a word is like striking a note on the keyboard of the imagination.
Remarks on Frazer's Golden Bough
An entire mythology is stored within our language.
Culture and Value
“When you are philosophizing you have to descend into primeval chaos and feel at home there.” 
Wisdom is passionless. But faith by contrast is what Kierkegaard calls a passion.
Ambition is the death of thought.
A man will be imprisoned in a room with a door that's unlocked and opens inwards; as long as it does not occur to him to pull rather than push it.
Reading the Socratic dialogues one has the feeling: what a frightful waste of time! What's the point of these arguments that prove nothing and clarify nothing?

Notebooks 1914-1916
Language is a part of our organism and no less complicated than it.
Certainly it is correct to say: Conscience is the voice of God.
What cannot be imagined cannot even be talked about.
What do I know about God and the purpose of life?I know that this world exists.That I am placed in it like my eye in its visual field.That something about it is problematic, which we call its meaning.This meaning does not lie in it but outside of it.That life is the world.That my will penetrates the world.That my will is good or evil.Therefore that good and evil are somehow connected with the meaning of the world.The meaning of life, i.e. the meaning of the world, we can call God.To pray is to think about the meaning of life.
To believe in a God means to understand the question about the meaning of life.To believe in a God means to see that the facts of the world are not the end of the matter.To believe in God means to see that life has a meaning.
Comparing Quotes
"Hell is other people" ~ No Exit, Jean-Paul Sartre 
"Hell is yourself and the only redemption is when a person puts himself aside to feel deeply for another person" ~ Tennessee Williams 
"(On Satre) Hell isn't other people. Hell is yourself" ~ Wittgenstein

Tractatus Logico-Philosophicus

Posted by Ali Reda | Posted in | Posted on 10/23/2013

Bertrand Russell’s Preface

As one with a long experience of the difficulties of logic and of the deceptiveness of theories which seem irrefutable, I find myself unable to be sure of the rightness of a theory, merely on the ground that I cannot see any point on which it is wrong. But to have constructed a theory of logic which is not at any point obviously wrong is to have achieved a work of extraordinary difficulty and importance. This merit, in my opinion, belongs to Mr Wittgenstein’s book, and makes it one which no serious philosopher can afford to neglect.


This book will perhaps only be understood by those who have themselves already thought the thoughts which are expressed in it or similar thoughts. It is therefore not a text-book. Its object would be attained if there were one person who read it with understanding and to whom it afforded pleasure. 
The book deals with the problems of philosophy and shows, as I believe, that the method of formulating these problems rests on the misunderstanding of the logic of our language. Its whole meaning could be summed up somewhat as follows: What can be said at all can be said clearly; and whereof one cannot speak thereof one must be silent.
 The book will, therefore, draw a limit to thinking, or rather—not to thinking, but to the expression of thoughts; for, in order to draw a limit to thinking we should have to be able to think both sides of this limit (we should therefore have to be able to think what cannot be thought). The limit can, therefore, only be drawn in language and what lies on the other side of the limit will be simply nonsense. 
How far my efforts agree with those of other philosophers I will not decide.Indeed what I have here written makes no claim to novelty in points of detail; and therefore I give no sources, because it is indifferent to me whether what I have thought has already been thought before me by another. On the other hand the truth of the thoughts communicated here seems to me unassailable and definitive. I am, therefore, of the opinion that the problems have in essentials been finally solved.


There are seven main propositions in the text. These are:
  1. The world is everything that is the case.
  2. What is the case (a fact) is the existence of states of affairs.
  3. A logical picture of facts is a thought.
  4. A thought is a proposition with a sense.
  5. A proposition is a truth-function of elementary propositions. (An elementary proposition is a truth-function of itself.)
  6. The general form of a proposition is the general form of a truth function, which is: [\bar p,\bar\xi, N(\bar\xi)]. This is the general form of a proposition.
  7. Whereof one cannot speak, thereof one must be silent.

 Questions of Ontology and Logical Atomism

World is the totality of all positive facts. Facts are made of states of affairs (atomic facts), Positive facts are the existence of states of affairs (what is the case) and from knowing what is the case, we can know what is not the case (negative facts); states of affairs are state of a combination of objects (objects and relations between them). Objects are simple, unanalyzable, and mutually independent. Facts exist in what Wittgenstein calls "logical space". Logical space is the domain of everything that is logically possible. For instance, though it is not true that Toronto is the capital of Canada, there is nothing illogical about supposing that it might be, so its possibility exists in logical space. Some items in logical space (for instance, "Ottawa is the capital of Canada") are true, while some items in logical space are false. True or false, everything in logical space is possible. "Love is purple" is not an item in logical space, because it is not logically possible (love is not the kind of thing to which we can ascribe a color).

Through Kenny's chess analogy, we can see Wittgenstein's logical atomism. For the sake of this analogy, the chess pieces (objects) and their positions (relations between objects) constitute game status (states of affairs) and therefore facts and the totality of facts is the entire particular game of chess. We can communicate such a game of chess in the exact way that Wittgenstein says a proposition represents the world. We might make a report for every piece's position using chess shortcuts like say "WR/KR1" to communicate a white rook's being on the square commonly labeled as king's rook 1. The logical form of our reports must be the same logical form of the chess pieces and their arrangement on the board in order to be meaningful. Our communication about the chess game must have as many possibilities for constituents and their arrangement as the game itself.

An object has internal and external properties. The internal properties are its logical form: what kind of object it is and how it can combine with\relate to other objects in states of affairs, it shows us which states of affairs the object can occur in. To return to the earlier example, we don't know what the number two is if we say it is purple. Thus, objects and their internal properties are what make up the substance of the world. But the material properties of the world are determined by the objects' external properties. For instance, the internal properties of yellow and red are indistinguishable (i.e they need light to be colored, so they are colorless internally without anything external) and they can both occur in the same sorts of states of affairs. The only way we can distinguish red from yellow is by their external properties, by saying that certain things are true of yellow that are not true of red, and vice versa. Substance is what subsists independently of what is the case.

Picture Theory of Representation

The logical picture of the facts is the thought. We can only think logically because thoughts must share the logical form of what they are about which are facts. Thoughts contain the possibility of the state of affairs which it thinks. We can imagine worlds other than this one, but we cannot imagine worlds that do not have the same logical form as this one. That is, we can imagine a world where horses speak and grass is pink, but we cannot imagine a world without space, time, or color. We can think contradictory things (e.g. "It is raining and it is not raining"), but we cannot think illogical thoughts that have no sense. I cannot think, "The number two is purple". A thought is a proposition with a sense\meaning. 
The sense of a proposition is its agreement and disagreement with possibilities of existence and non-existence of states of affairs.
A propositional sign is any form of communication by a perceptible sign (written, spoken or signs) used in transmitting thoughts. A proposition is a propositional sign’s projective relation to the world. The combination of objects in the state of affairs corresponds to the combination of objects of the thoughts corresponds to the combination of simple signs in the propositional sign. All share the same form and logical structure. The simple sign is the name of the object. The name means the object. The object is its meaning.
To understand a proposition means to know what is the case if it is true. It is understood by anyone who understands its constituents.
To reach the level of the state of affairs and its objects, we analyze the complex objects of everyday speech into simpler parts by means of definitions. In the analysis of "Plato talks with his pupils", "Plato" needs to be replaced with something like "the man who was the teacher of Aristotle". Wittgenstein went further and argued that it must be possible to continue this kind of analysis to a point at which no more subdivision would be possible. Ultimately, simple symbols—the symbols for names—cannot be further defined: they are fully analyzed so we cannot say what they mean.  The meaning of a name is external to the name. The meaning of a name is the object it denotes, and there is nothing in the name itself (as a written or spoken sign) that can tell us what object it denotes. Rather, we learn the meaning of a name by observing how and in reference to what it is used. The meaning of a name lies outside the name. We must show what they mean by using what Wittgenstein calls "elucidations”. They are propositions which contain the names. 

Propositions make "pictures" of facts in the world. The picture is a model of reality; it represents a fact by having its logical structure. The logical structure of the picture is the relation between the elements of the picture- the internal properties of these objects-. The logical structure of the fact is its states of affairs, the relation between its objects. I.e. The elements of a picture correspond to the objects of a fact. The situation represented by a picture is the sense of the picture. Comparing the picture of a fact and the reality – the sense of the picture agrees with reality or not- shows the truthfulness of a fact in the world. 
It is laid against reality like a measure.
This comparison implies that the truthfulness of a fact requires empirical evidence. But this empirical experience is subjective, i.e it depends on the subject's beliefs, sensory organs. like one person who sees illusions and another one who doesn't.

Wittgenstein calls the possibility that things are related to one another -the possibility of this logical structure- "pictorial form". 
A picture cannot, depict (describe) its pictorial form: it displays it.
He is making the important distinction between saying and showing. 
What can be shown cannot be said.
In this way, linguistic expression can be seen as a form of geometric projection, where the many languages are the different forms of projection but the logical structure of the expression is the unchanging geometric relationships. We cannot say with language what is common in the structures, rather it must be shown, because any language we use will also rely on this relationship, and so we cannot step out of our language with language and in general stepping out of language is stepping out of our thoughts or out of the logical space which is impossible. And it is only because of this commonality that speech can be understood. 
What expresses itself in language, we cannot express by means of language.
The sense of a proposition is internal to the proposition. While propositions can depict all of reality, they cannot depict its logical form.
A proposition shows its sense. A proposition shows how things stand if it is true. And it says that they do so stand.
Wittgenstein was inspired for this theory by the way that traffic courts in Paris reenact automobile accidents.  A toy car is a representation of a real car, a toy truck is a representation of a real truck, and dolls are representations of people. In order to convey to a judge what happened in an automobile accident, someone in the courtroom might place the toy cars in a position like the position the real cars were in, and move them in the ways that the real cars moved. In this way, the elements of the picture (the toy cars) are in spatial relation to one another, and this relation itself pictures the spatial relation between the real cars in the automobile accident.


Every part of a proposition which characterizes its sense is an expression (a symbol). Wittgenstein accepts Frege's and Russell's view of a proposition as a function of the expressions. For instance, the variable function "the x is on the y." only gives sense when propositional variables are filled with expressions such as "hat" or "table" thus the variable function becomes a proposition.

The expressions themselves are meaningless outside the context of a proposition. This is an important remark to show that Language is the Source of Philosophical Confusion, when philosophers get confused over questions like whether the Good is more or less identical than the Beautiful or of what knowledge is; they are not confused because the essence of knowledge is difficult to identify. Rather, they are confused because they have abstracted a word from the contexts in which it has a function and find that, outside these contexts, the word loses its meaning. Another source of confusion is in the language of everyday life it very often happens that the same word signifies in two different ways—and therefore belongs to two different symbols. For example the word “is” appears as the copula, as the sign of equality, and as the expression of existence. Another source of confusion is that two words, which signify in different ways, are apparently applied in the same way in the proposition. For example, in the proposition “Green is green”—where the first word is a proper name as the last an adjective.
It is not surprising that the deepest problems are in fact not problems at all.
In order to avoid these errors, we must employ a symbolism which excludes them, by not applying the same sign in different symbols and by not applying a sign in the same way which signify different meanings.  Two functions being used in different ways cannot possibly have the same meaning, i.e. be the same function.  So no proposition can make a statement about itself, because a propositional sign cannot be contained in itself or else it would signify two different meanings.

Wittgenstein criticizes Russell's Theory of Types saying it can be disposed of simply by recognizing that a proposition that makes a statement about itself is being used in two different ways, and so it cannot be the same proposition. The "F" in "F(fx)" and the first "F" in "F(F(fx))" range over different kinds of variables, the letter by itself signifies nothing. He is saying that the theory of types is implicit in logic and a priori, Russell’s' paradox shouldn't happen in the first place. 
The rules of logical syntax must go without saying, once we know how each individual sign signifies.
Wittgenstein introduces the notion of formal, or internal, properties, those properties that show themselves (as opposed to being spoken about) in a proposition. These properties define the logical structure of propositions, facts, and objects. We can’t think of the object as not having them. It would be as nonsensical (illogical) to ascribe a formal property to a proposition as to deny it the formal property. Propositions have the same internal properties as the facts they depict.

Ferge said that any proposition of the form "x is a y," "x" will represent an object and "y" will represent a concept. According to Wittgenstein, there is a fundamental difference between "x is a horse" and "x is a concept." Only grammar leads us to think the two are equivalent. A formal concept defines the formal properties of an object, state of affairs, or fact. Like "x is a number" We cannot say that x is a number: it being a number shows itself. Any attempt to use a formal concept in a proposition (e.g. "two is a number," "purple is a color") will result in a nonsensical pseudo-proposition. A concept proper is a function and can feature in propositions, like "x is a horse”.

Logic and Propostions as Truth Functions

Propositions are built up as truth-functions of elementary propositions. An elementary proposition is a truth-function of itself. A proposition that shares all the truth- grounds of one or several other propositions is said to follow from those propositions. If one proposition follows from another, we can say the sense of the former is contained in the sense of the latter. For instance, the truth-grounds for "p" are contained in the truth-grounds for "p.q" ("p" is true in all those cases where "p.q" is true), so we can say that "p" follows from "p.q" and that the sense of "p" is contained in the sense of "p.q."

Wittgenstein emphasizes that the truth-value of a proposition has no bearing on its sense. True or false, it still makes a picture of the world, and we can still draw logical inferences from that picture. The propositions p and ~p depict the same possible situation, only they have opposite sense one says that the picture presented is the case, and the other says that it is not the case.

Wittgenstein's work here is that the sense of a proposition is given if its truth conditions are given. If we know under what circumstances a proposition is true and under what circumstances it is false, then we know all there is to know about that proposition. This is exactly what truth tables do. Any proposition, according to Wittgenstein, consists of one or more elementary proposition, each of which can be true or false independently of any other. If we put all the elementary propositions that constitute a given proposition into a truth table that lists all the possible combinations of true or false that can hold between them, we will have an exhaustive list of the truth-conditions of the given proposition. Thus, a truth table can show us the sense of the proposition. In logic process and result are equivalent. Wittgenstein did not invent truth tables, but their use in modern logic is usually traced to his introduction of them in the Tractatus.

The great advantage of this notation is that it expresses the sense of a proposition without any of the connectives (Logical constants) we normally find in logical notation, such as "and," "or," and "if… then." Frege builds his entire system from the "primitive" connective "not" and "if…then." Russell builds his from "not" and "or." These "primitive" connectives are in fact interchangeable (Frege's "if p then q" can be expressed in Russell's system as "q or not p," and Russell's "p or q" can be expressed by Frege's "if not p then q"). If the same proposition can be expressed in a handful of different ways then clearly, none of these connectives are essential to the sense of the proposition, thus giving credence to Wittgenstein's "fundamental idea" that "the ‘logical constants’ are not representatives; that there can be no representatives of the logic of facts". Since all these propositions and axioms are equivalent so there are no differences between them, so as a tautology says nothing, all the propositions of logic say the same thing: nothing. Wittgenstein is trying to dissociate the importance of notation from logic itself. In a truth table, the connections between elementary propositions "show" themselves, and so need not be said.
Even the negation that occurs in a proposition, is no characteristic of its sense because if (ssp = p) then denial is already contained in affirmation. And if there was an object called “s”, then “ssp” would have to say something other than “p”.
Logic must look after itself. What makes logic a priori is the impossibility of illogical thought.
In this statement, he is alluding to a further difference between his conception of logic and the Universalist conception espoused by Frege and Russell. According to the Universalist conception, certain logical axioms must be laid out as fundamental "laws" of logic. Wittgenstein explains that his method can "show" the workings of logical inference, thus rendering unnecessary the "laws of inference" that both Frege and Russell had built into their axiomatic systems. One proposition follows from a second proposition if the first is true whenever the second is true. If we express "p or q" as "(TTTF)(p,q)" and "p and q" as "(TFFF)(p,q)" we can see that the former follows from the latter by comparing their truth-grounds: where there is a "T" in the latter proposition, there is a corresponding "T" in the former proposition. We don't need a law of inference to tell us this: it shows itself plainly in the truth-grounds of the two propositions. We should not need external laws to tell us how proceed with logic since there is nothing external to logic. Logic should not stand in need of justification.
Hence, there can never be surprises in logic.
His conception of logic is explained: 
"The propositions of logic describe the scaffolding of the world, or rather they represent it." 
The logic of the world which the propositions of logic show in tautologies, mathematics shows in equations.
Logic is the common structure that links our minds to the universe but what is its relation to mathematics?
The metaphor of scaffolding brings to light four principal aspects of Wittgenstein's conception of logic. First, scaffolding is a framework structure: it is a skeleton of joints rather than a building with walls and rooms. Similarly, logic does not consist of propositions with a sense, but only provides a framework within which propositions with a sense may fit. Second, the framework of scaffolding is used to construct a more substantial building, just as logic provides a framework within which the substantial facts about the world may fit. Third, scaffolding has points of contact with the building it is placed against, but it does not overlap with the building, nor is it a part of the building. Logic has points of contact with the world in that both logic and the world share a logical form, but the content (as opposed to the form) of facts themselves has no analogue in logic. Fourth, scaffolding is only a tool used in construction: a sturdy and complete building has no need of scaffolding. Similarly, we do not need logic or philosophy when language is functioning normally. These tools are only needed to provide clarity when language misfires and attempts to speak nonsense.

The “experience” which we need to understand logic is not that such and such is the case, but that something is; but that is no experience. Logic precedes every experience—that something is so. It is before the how, not before the what.

A proposition that is true no matter what (e.g. "(TTTT)(p,q)") is called a "tautology" and a proposition that is false no matter what (e.g. "(FFFF)(p,q)") is called a "contradiction". Tautologies and contradictions lack sense in that they do not represent any possible situations (state of affairs) so they are senseless but they are not nonsense either, because they consist of elementary propositions and are held together in a logical way. It is clear that the logical product of two elementary propositions can neither be a tautology nor a contradiction. The assertion that a point in the visual field has two different colours at the same time is a contradiction.

The operation is that which must happen to a proposition in order to make another out of it. Like combining the two propositions, "p" and "q," to form a new proposition "p.q". An operation combining elementary propositions in a truth- function is a truth-operation. An operation is not a form or object in its own right; it simply expresses the difference between the forms of two propositions or in other words the relation that stands between the structure of the base proposition and the structure of the resulting proposition.

Wittgenstein attempts to rid logical notation of the signs for generality and identity. Whenever a variable is given, that variable expresses all objects that can take that variable place, so generality is already given when a variable is given. We don't need an additional sign to denote generality. As for identity, to say of two things that they are identical is senseless because they can’t have two different signs referring to one meaning and to say of one thing that it is identical with itself is to say nothing at all.

A proposition of the form "A believes that p" does not actually involve a relationship between A and the proposition "p". For A to think, believe, or say that p is the case. They relate p to the verbal expression of p, so that what we are really saying is "'p' says that p". And the internal similarity between the words and the proposition is obvious. Wittgenstein further infers that there is no such thing as a "soul" where thoughts and beliefs reside.

Wittgenstein gives a general form for expressing a term in a particular series as "[a, x, O'x]," where "a" stands for the first term in the series, "x" stands for an arbitrarily selected term, and "O'x" stands for the term that immediately follows "x." The "O'" is the operation by which a term in the series is generated out of another. So, for instance, we could express the series of square numbers as [1, x,(sqr(x) + one)^2].
Wittgenstein takes the successive application of an operation as the model of a proposition. His definition of the general propositional form as "[?p, ??, N(??)]" is a variation of the general form for expressing a term in a series: "[a, x, O'x]." The "?p" is the collection of elementary propositions that a given proposition is composed of, and thus is the first term in the series of operations that generates a complex operation. The "??" is a complex proposition in this series of successive negations, and "N(??)" shows us how the next term in the series will be generated, namely by negating all the terms in "??."

Wittgenstein observes that all propositions can be derived by means of successive applications of the operation (——T)(?,….), that is by means of negating all the terms in the right hand pair of brackets. When Wittgenstein claims that all propositions can be derived by successive applications of a negating operation, he is alluding to the "Sheffer stroke," a logical constant discovered in the early 20th century. While Frege develops a system that relies only on the logical constants "not" and "if…then," and Russell develops a system that relies only on the logical constants "not" and "or," it was discovered that the Sheffer stroke—usually symbolized as a vertical bar, "|"—was a logical constant that could stand on its own. The proposition "p|q" is equivalent to "~p.~q." Thus, "~p" can be expressed "p|p," "p v q" can be expressed "(p|q)|(p|q)," and so on. Wittgenstein draws on the Sheffer stroke to show that a single operation can be used to derive any proposition from any other proposition. Given the elementary propositions, we can generate all other propositions using the Sheffer stroke.

Frege's search for something more certain than pure intuition to ground the concepts of was largely against Kant, who argued that our knowledge of mathematics is based on pure intuition. Any given number could be generated, according to Kant, by adding a certain number of ones: 4 = 1 + 1 + 1 + 1, while 98 = 1 + 1 + 1 + …. Pure intuition is necessary for the concept of "and so on" that makes it possible to add infinitely many ones together. Frege claimed that he could make pure intuition unnecessary to mathematics by giving a definition of number based in logic that would provide a general rule more rigorous than "and so on" for adding successive ones. Frege and Russell both developed ingenious systems to prove that the laws of mathematics could be inferred from basic logical axioms, they were largely successful.

In defining mathematics as a "method of logic", Wittgenstein suggests that numbers are not objects that can be constructed out of logical forms. Numbers are exponents of operations: they constitute an index for expressing how many times an operation has been applied. Thus, the propositions of mathematics do not say anything about the world, but only reflect the method in which propositions are constructed. So the logic of the world which the propositions of logic show in tautologies, mathematics shows in equations. Mathematical propositions express no thoughts. The propositions of mathematics are equations, and therefore pseudo-propositions. Because they contain properties of affirmation; two expressions may have same meaning but both are used unlike logic. It is a property of “1+1+1+1” that it can be conceived as “(1+1)+(1+1)”. The equation characterizes only the standpoint from which I consider the two expressions, that is to say the standpoint of their equality of meaning. The curious thing about Wittgenstein's philosophy of mathematics in the Tractatus is that it relies on the concept of "and so on" that Frege had gone to such lengths to eliminate.

Language and Solipsism 

The limits of my language  mean the limits of my world. Logic fills the world: the limits of the world are also its limits. 
The language which I understand, is the representation of my world, we cannot therefore say in logic: This and this there is in the world, that there is not. For that would apparently presuppose that we exclude certain possibilities and this cannot be the case since otherwise logic must get outside the limits of the world. 
What we cannot think, that we cannot think: we cannot therefore say what we cannot think.
This observation leads Wittgenstein to reflect on the limited truth of solipsism. The term "solipsism" defines a number of related philosophical positions, all of which claim that the objects and people in the world only exist as objects of my awareness, that only I, as a thinking consciousness, truly exist. Wittgenstein draws the analogy between the relationship between the metaphysical subject and the world on one hand, and the relationship between the eye and the visual field on the other. I cannot see my eye anywhere in my visual field, but the existence of a visual field presupposes the existence of the eye. Similarly, myself is not something I encounter in the world, but my experience of the world presupposes that there is a self to experience it.

However, I cannot talk about this self because it is outside the limits of the world, and hence outside the limits of language. The philosophical I is not the man, not the human body or the human soul of which psychology treats, but the metaphysical subject, the limit—not a part of the world much as the eye is the limit of the visual field. So there are no objects or elementary propositions that correspond to this "I": there are no propositions with sense, true or false, relating to it.

So both language and the world share the same limits leads to the reflection that solipsism is correct in the claim that "the world is my world. However, the thesis of solipsism cannot be put into language, but can only show itself. At this point, however, there is no real distinction between solipsism and pure realism, The I in solipsism shrinks to an extensionless point and there remains the reality co-ordinated with it.

Logic and Laws of Nature

Logic determines the form that laws of nature can take, but it does not it make any claims regarding nature. Scientific laws themselves do not belong to logic, because they make claims about experience and do not hold a priori. The law of induction, the law of causality, and other such scientific principles are not exactly empirical facts, either. 
All such propositions, including the principle of sufficient reason, the laws of continuity in nature and of least effort in nature, etc. etc.—all these are a priori insights about the forms in which the propositions of science can be cast.
They define the framework within which we can talk about natural phenomena. It is clear that there are no grounds for believing that the simplest course of events will really happen. 
That the sun will rise tomorrow is a hypothesis; and that means that we do not know whether it will rise. A necessity for one thing to happen because another has happened does not exist. There is only logical necessity.
Wittgenstein when discussing the concept of causality, that to comprehend it we need to understand the two actions in sequence as relative to each other and not a law of causality itslef. He gives an example on measuring time, we cannot compare any process with the “passage of time”—there is no such thing—but only with another process (say, with the movement of the chronometer). Hence we can describe the lapse of time only by relying on some other process. Time is proved as a dimension after the special relativity which renders his example a fault, none the less he is correct.

The Kantian problem of the right and left hand which cannot be made to cover one another already exists in the plane, and even in one-dimensional space; where the two congruent figures a and b cannot be made to cover one another without moving them out of this space. The right and left hand are in fact completely congruent. And the fact that they cannot be made to cover one another has nothing to do with it. A right-hand glove could be put on a left hand if it could be turned round in four-dimensional space.

Wittgenstein compares the laws of nature to a square mesh laid out over a surface of black and white spots. This mesh allows us describe the surface by saying of each square in the mesh whether it is black or white. Of course, a triangular mesh or a hexagonal mesh could be used just as well as a square mesh, though certain kinds of mesh will likely provide a simpler and more accurate description of the surface than others. And while the mesh itself can tell us nothing about the distribution of black and white on the surface, we can learn about the surface by observing what kinds of mesh describe it most accurately. The laws of nature do not tell us anything about the world, nor are they necessarily true of the world. Rather, they are tools we can use to make sense of the world in order to understand its regularities with greater clarity in order to be able to describe reality. Science is ultimately descriptive, not explanatory.

Answerable Questions and the Mystical

Questions can only be answered when the questions themselves can be framed in words. Thus, we can only ask questions and get answers regarding facts about the world, and not about anything transcendent. 
We feel that even when all possible scientific questions have been answered, the problems of life remain completely untouched. Of course there are then no questions left, and this itself is the answer.
For an answer which cannot be expressed, the question too cannot be expressed. The riddle does not exist. If a question can be put at all, then it can also be answered. 
Skepticism is not irrefutable, but palpably senseless, if it would doubt where a question cannot be asked. For doubt can only exist where there is a question; a question only where there is an answer, and this only where something can be said.
Specifically, Wittgenstein limits propositions to making claims, true or false, about how things stand in the world, which is the business of natural science. To think of philosophy as made up of propositions is a common error. The role of philosophy is not to state truths about the universe, but to clarify the truths stated by the other sciences. The business of philosophy is not of saying, but of showing: philosophy clarifies the logical structure of our propositions that is clouded by everyday language. Philosophy is an "activity" and not one of the natural sciences. Since philosophy cannot step outside the boundaries of language, it should act as a watchdog at those boundaries, clarifying vague propositions. He concludes that the only correct method in philosophy is to confine oneself to what can be spoken, and, whenever others try to say the unsayable (ethics, aesthetics, metaphysics, etc.), to point out to them that they are speaking nonsense. Wittgenstein asserts that most philosophical confusion arises from trying to speak about things that can only be shown. 
Without philosophy thoughts are, as it were, cloudy and indistinct: its task is to make them clear and to give them sharp boundaries. 
Philosophy limits the disputable sphere of natural science. It should limit the thinkable and thereby the unthinkable. It should limit the unthinkable from within through the thinkable. It will mean the unspeakable by clearly displaying the speakable. Everything that can be thought at all can be thought clearly. Everything that can be said can be said clearly.
Anything transcendental can’t be said because there is no perspective external to the world from which we can talk about the world or its contents generally. For Wittgenstein, ethical `propositions' are absolute judgments of value of the form, ethics pervades all of life; Ethics are transcendental because no aspect of life is untouched by ethics. Our attitude toward the world shapes the world we live in. Thus, we cannot talk about ethics since logical language can only reflect the world, any discussion of the mystical that which lies outside of the metaphysical subject's world, is meaningless. Actions are not good or bad because of their consequences, but because of the overall attitude toward life that they embody. While the exercise of the will has no direct effect on the world itself, this exercise of the will defines the kind of world a person inhabits: 
"The world of the happy man is a different one from that of the unhappy man".
Next he talks about the value of the world and its meaning. Because all propositions are of equal value in the world so they cannot express anything higher. Accidental is what isn't necessary. He concludes:
The sense of the world must lie outside the world. In it there is no value—and if there were, it would be of no value.If there is a value which is of value, it must lie outside all happening and being-so. For all happening and being-so is accidental. What makes it non-accidental cannot lie in the world, for otherwise this would again be accidental. It must lie outside the world.
The Sensible can be said because it is logical, It can be thought of and it is possible in our world. They can be pictured as the elements in a picture. It's truthfulness can be confirmed by comparing with the real world so it can be true or false. 

What can be shown and not said it what is outside language and thought, what can't be defined, what can't be thought about, like the logical form of the world or elucidations, they are tautologies so they are senseless. It is the relation between the pictures' elements so It can't be described by any picture but can only be shown in a picture. 
‘A state of affairs is thinkable’: what this means is that we can picture it to ourselves.
What can't be be put into words, (inexpressible so can neither be said nor shown)is what lies out of the world, is illogical or nonsensical, it also what can't be defined, what can't be thought about and what can't be pictured  nor be shown in a picture, because there is no mapping between elements of the picture and the objects. For example God has no propositions and its components can't be defined so they can't be put into words or be thought about so they are nonsensical.

Then he elaborates on the mystical:
There are, indeed, things that cannot be put into words. They make themselves manifest. They are what is mystical. It is not how things are in the world that is mystical, but that it exists. Feeling the world as a limited whole—it is this that is mystical.


The controversial ending proposition of the book:
My propositions are elucidatory in this way: he who understands me finally recognizes them as senseless, when he has climbed out through them, on them, over them. (He must so to speak throw away the ladder, after he has climbed up on it.) He must surmount these propositions; then he sees the world rightly.
There are, at present, two dominant ways to read Wittgenstein's Tractatus Logico-Philosophicus (TLP). 

  1. The irresolute reading takes what is called a view of nonsense: it takes Wittgenstein's propositions in the TLP to be nonsensical in that they are trying to express what, according to Wittgenstein, can only be shown, e.g. propositions about the logical form of propositions. The traditional interpretation, perhaps best represented by P. M. S. Hacker, takes Wittgenstein to be pointing out that the kinds of subject matter he treats of lies outside the realm of sensible discourse. The Tractatus treats of things that cannot be said, but can only be shown.
  2. The resolute reading on the other hand takes what is called an austere view of nonsense: this takes the propositions in the TLP to be actual, irredeemable nonsense. In this sense the whole of TLP becomes a quasi-ironic argument against transcendental idealism. The locus classicus of this reading is Cora Diamond's The Realistic Spirit. 

As a result, several distinctions can be made between the two readings. For example, if we are irresolute then we take Wittgenstein to be a realist, whereas if we are resolute the question of realism/anti-realism does not arise. 

However, we can try to understand the frame of mind that would be necessary to think that these propositions make sense. They are evident as logic itself. Wittgenstein is not trying to tell us a number of things that we did not already know; he is trying to instruct us in a way of thinking that will help us out of philosophical muddles. While the propositions of the Tractatus may themselves be nonsense, Wittgenstein hopes that they have served their instructive purpose. We are expected to put down this book not with the knowledge that the world is made up of objects and states of affairs and those propositions depict facts, but with an understanding of why it is impossible to say these sorts of things. The goal of the Tractatus, as Wittgenstein claims in his preface, is "to draw a limit … to the expression of thoughts."

Wittgenstein concludes:
"What we cannot speak about we must pass over in silence".