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Brief Lives

Gottlob Frege (1848-1925)

Alistair MacFarlane laments the profound disappointment of a master of logic.

Nothing in our intellectual life seems more secure than arithmetic and logic. When we buy something, hand over cash and receive change, the statement: “This amount of change is correct” must be true or false. This is so despite the country, the currency, the article purchased, the market involved, and a myriad of other factors.

Logic and arithmetic seem to fit together naturally. Imagine then the colossal shock felt by Gottlob Frege, who had spent years attempting to prove that arithmetic could be based on logic, when he received a letter from Bertrand Russell demonstrating that his proof was fallacious. Something momentous had happened: Frege had collided with the limits of logic. It was as significant for philosophy as Max Planck’s discovery that energy could not be arbitrarily sub-divided was for physics.

Gottlob Frege by Gail Campbell, 2016

A Life in Logic

Friedrich Ludwig Gottlob Frege, the central figure in one of the most dramatic events in the history of philosophy, was born on 8th November 1848 in Wismar on the Baltic coast of Germany. His father, Karl Alexander Frege, was headmaster of a high school for girls that he had founded. His Polish mother, Auguste Wilhelmine Sophie (née Bialloblotzky), taught in it. In 1866, Karl Frege died and Gottlob’s mother took the school over, enabling young Frege to complete his secondary education and enter the University of Jena in 1869. Mother and son became, and remained, very close. After he went to university, she sold the school and moved to live with him.

Frege studied mathematics and physics before moving to the University of Göttingen, where in 1873 he gained a doctorate in mathematics. Returning to Jena, he joined the staff of the mathematics department of the University, remaining there for the rest of his life. His sole form of relaxation was long walking trips. On one of these he met Margarete Liseberg, and they were married in 1887. They had no children, but after her death he adopted a six-year-old, Alfred, and brought him up.

Frege’s doctoral thesis was on projective geometry. Thus the man who many say became the greatest of all logicians started his professional life as a geometer. What led him to logic? Frege was a deep and tenacious thinker, and his study of the new forms of non-Euclidean geometry which were beginning to emerge had led him to ask himself a profound question: If the sublime edifice of geometry is built on solid logical foundations, why is this not the case for the rest of mathematics? Specifically, why is this not the case for arithmetic? Since its emergence in ancient Greece, classical geometry had been a logically deduced system based on Euclid’s famous set of axioms and postulates. What’s different about arithmetic? Frege spent the rest of his life seeking an answer.

To use a system of deductive logic for any purpose, we must first do two things:

• Define the entities to be discussed. (In Euclid’s case these were lines and points.)

• Generate an appropriate set of axioms. (In Euclid’s case, that two lines intersecting define a point; two points joined define a line, and so on.)

The set of entities should be minimal, yet complete for the purpose. A set of axioms should be sufficient for the discussion, compatible, and coherent.

Frege soon realised that no form of logic existed which was sufficiently powerful for his intended purpose of creating a logical foundation for arithmetic, so he made the heroic decision to create one. Single-handedly, he developed what is now modern logic. This supreme achievement can only be compared with Newton’s creation of the calculus or Einstein’s creation of relativity theory. The triumph and tragedy of Frege’s life is that, having created a new and more powerful form of logic, he was unable to create an appropriate set of axioms for arithmetic.

Frege began the quest in his book Begriffschrift (1879); the name means Concept Script. In it he developed what is now called ‘propositional calculus’, which deals with those logical inferences which depend on applying negation, conjunction, or disjunction to whole sentences. That is, it seeks to establish the truth or falsehood of composite sentences joined together through such logical operations. The truth values of these composite sentences are fixed by the truth values of their component parts. For instance, if ‘the sun is shining’ is true, then ‘not (the sun is shining)’ is false. Frege also introduced an unprecedented degree of precision into logic by introducing existential and universal quantifiers, codifying the logically useful statements, ‘There is an x such that…’ and ‘For all xs…’. He then set about the great task of attempting to establish arithmetic, and hence mathematics, on a basis of logic.

Out For The Count

The crucial step was to decide how to characterise numbers, and thence devise a suitable set of axioms for arithmetic. This led him, like Russell (see Brief Lives, PN 89) and Whitehead (PN 86), to define numbers in terms of classes. Thus the number 2 would be the class of all pairs, 3 would be the class of all trios, and so on. On this basis Frege drew up what he considered a suitable set of axioms and proceeded to write his monumental Grundgesetze der Arithmetik (Foundations of Arithmetic), publishing the first volume in 1884.

He was correcting the proofs of the second volume when he received a letter from Russell, who was carefully going through the first. Russell had found a paradox that would wreck Frege’s entire project if it could not be resolved. On reading the fateful letter and grasping its implications, Frege was utterly downcast. He replied, thanking Russell with great courtesy and appreciation, but revealed his dismay by saying “Arithmetic totters!”

It was in fact a blow from which Frege’s project never recovered. After drawing attention to the problem, and in vain attempting to repair the damage, he went ahead and published his remaining volume anyway. Later writing about this episode, Russell said, “As I think about acts of integrity and grace, I realise that there is nothing in my knowledge to compare to Frege’s dedication to truth. His entire life’s work was on the verge of completion, much of his work had been ignored to the benefit of men infinitely less capable, his second volume was about to be published, and upon finding that his fundamental assumption was in error, he responded with intellectual pleasure, clearly submerging any feelings of disappointment. It was almost superhuman, and a telling indication of that of which men are capable if their dedication is to creative work and knowledge instead of cruder efforts to dominate and be known.” (Frege To Gödel: A Source Book In Mathematical Logic, 1967, p.127.)

Frege published little more in the remaining years of his life. When he made his will in 1925 he left all his unpublished papers to Alfred, with a letter in which he wrote, “Do not despise the pieces I have written. For even if it is not all gold, there is gold in them. I believe there are things which one day will be prized more highly than they are now. Take care that nothing is lost.”

He died six months later, on 26 July 1925, in Bad Kleiner near his birthplace in Wismar, now part of the Nordwestmecklenburg district on the north bank of the Schweriner See. His death was scarcely noticed by the world. But nothing was lost, and Frege is now regarded as one of the greatest of all logicians, and as having had a profound effect on philosophy.

What Went Wrong?

Frege had defined numbers in terms of classes. The difficulty that ruined his project is that while some classes may be members of themselves, other closely related classes may not. The source of the problem can be easily understood by considering the following two statements:

• The class of integers cannot be a member of itself for the simple reason that a class is not the same thing as an integer.

• However, the class of non-integers is a member of itself for the very same reason. (A class is a non-integer!)

Such deceptively simple, but fundamental, properties of classes or sets trip up all but the super-cautious and the logically meticulous. The unfortunate result, for Frege, was that some deductions made from the axiom set used in his Grundgesetze would be neither true nor false, thus vitiating logical argument. The true impossibility of Frege’s project of grounding mathematics on logic only emerged after his death when, in 1931, Kurt Gödel published his Incompleteness Theorem (see PN 91). This Theorem logically proved the necessary incompleteness of any attempt to logically systematize arithmetic.

Philosophy Post-Frege

Einstein made an unforgettable remark when a visitor introduced himself as a philosopher of religion. “Ach,” said Einstein “das muss ein eiertanzen sein!”: “That must be like dancing on eggs!” The only people who need to worry about the aftermath of Frege’s calamitous setback are philosophers of mathematics. The rest of us can follow the advice of the greatest post-Fregean philosopher, Ludwig Wittgenstein (PN 87): If seeking the meaning, look for the use. Or, numbers are what numbers do. But the spectacular failure of the attempt to reduce arithmetic to logic raises wider questions about the role of logic in philosophy, as well as about the difference between philosophy and science.

Willard Quine (PN 95) believed that logic is all we’ve got and that better forms of logic are all we’re ever likely to have, and concluded that philosophy could now only be an integrating framework for science. Hilary Putnam (PN 114) disagreed profoundly. In the last sentence of the last lecture he gave to his Harvard students, Putnam told them why philosophy remains a distinct and worthwhile pursuit. Engaging in philosophy, he said, means that you take the responsibility of always trying “to think deeply and with integrity.” Before Frege’s disappointment this would have been a truism. After Frege, it contains a deeper truth, since thinking deeply must mean much more than thinking logically. Thinking deeply must involve imaginative new ways of thinking, and this will place severe demands on intellectual integrity.

Putnam is a quintessential post-Fregean philosopher, and a careful consideration of his work shows some of the ways in which philosophy might develop. One is to use thought experiments, of which Putnam’s most famous are the ‘Brains in a Vat’ and the ‘Twin Earth’. ‘Brains in a Vat’ first appeared in the opening chapter of Putnam’s Reason, Truth and History (1981), achieving instant notoriety and forming the basis for the film The Matrix. The argument is a modern version of Descartes’ evil demon. Adopting a mood of scepticism, we could be no more than bodiless brains nourished in a vat and supplied with information simulating an external world. Putnam argues that this situation is incoherent; that we must be more than merely perception machines. He argues for externalism – for the necessity of external factors being crucial for us to know and think coherently about a world that contains us. For instance, arithmetic is about the discrete finite objects with which we interact in the physical world.

The ‘Twin Earth’ argument is given in Putnam’s best-known work, The Meaning of ‘Meaning’ (1975). Here he asks us to imagine a facsimile of our Earth which is similar to ours in every respect save one. The sole difference is that the water on Twin Earth, although looking, feeling and tasting like ours, and filling rivers, lakes and oceans like ours, has a different chemical composition. Thus if one were to visit Twin Earth, when using the word ‘water’, we would be referring to a different thing. Since, by definition, we would be identical in every other way, including our thought processes, Putnam concludes that meaning cannot simply be a function of our internal mental processes. In this way, the meaning of numbers arises from the way we use them in our daily lives.

Frege’s Legacy

Frege’s failure to prove that arithmetic was no more than logic was one of the defining moments in philosophy. It has clarified philosophy’s independence from science. Science is about facts obtained by careful experiments, on the results of which are based explanatory theories about the physical world. Philosophy is about thinking, deeply, and subject to no constraints other than the need to preserve intellectual integrity.

Sir Alistair MacFarlane is a former Vice-President of the Royal Society and a retired university Vice-Chancellor.