Childhood —69 [ edit ] Frege was born in in Wismar , Mecklenburg-Schwerin today part of Mecklenburg-Vorpommern. Frege studied at a grammar school in Wismar and graduated in Studies at University —74 [ edit ] Frege matriculated at the University of Jena in the spring of as a citizen of the North German Confederation. In the four semesters of his studies he attended approximately twenty courses of lectures, most of them on mathematics and physics. His most important teacher was Ernst Karl Abbe —; physicist, mathematician, and inventor. Abbe gave lectures on theory of gravity, galvanism and electrodynamics, complex analysis theory of functions of a complex variable, applications of physics, selected divisions of mechanics, and mechanics of solids.

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Indeed, for each condition defined above, the concepts that satisfy the condition are all pairwise equinumerous to one another. This extension contains all the concepts that satisfy Condition 0 above, and so the number of all such concepts is 0. Frege, however, had a deep idea about how to do this. Note that the last conjunct is true because there is exactly 1 object namely, Bertrand Russell that falls under the concept author of Principia Mathematica other than Whitehead.

Using this definition as a basis, Frege later derived many important theorems of number theory. Philosophers appreciated the importance of this work only relatively recently C.

Parsons , Smiley , Wright , and Boolos , , It was recently shown by R. Their different conceptions of logic helps to explain why these two philosophers came to such different conclusions. In particular: What resources or laws did Kant and Frege both consider to be logical? Did Kant and Frege agree about the content and subject matter of logic? An answer to the first question sets the stage for answering the second. The debate over which resources do and do not require an appeal to intuition is an important one.

Frege continued a trend started by Bolzano , who eliminated the appeal to intuition in the proof of the Intermediate Value Theorem in the calculus which in its simplest form asserts that a continuous function having both positive and negative values must cross the origin. Bolzano proved this theorem from the definition of continuity, which had recently been given in terms similar to the definition of a limit see Coffa , But appeal to a graph involves an appeal to intuition, and both Bolzano and Frege saw such appeals to intuition as potentially introducing logical gaps into a proof.

Frege dedicated himself to the idea of eliminating appeals to intuition in the proofs of the basic propositions of arithmetic. The Rule of Substitution allows one to substitute complex formulas for free second-order variables in logical theorems to produce new logical theorems.

Thus, the Comprehension Principle for Concepts asserts the existence of a concept corresponding to every expressible condition on objects. And Kant takes the laws of logic to be normative and prescriptive something one can get wrong , and not just descriptive , 16 ; they provide constitutive norms of thought MacFarlane , 35; Tolley By contrast, Frege rejects the idea that logic is a purely formal enterprise MacFarlane , 29; Linnebo , He took logic to have its own unique subject matter, which included not only facts about concepts concerning negation, subsumption, etc.

Frege , [, ] says: Just as the concept point belongs to geometry, so logic, too, has its own concepts and relations; and it is only in virtue of this that it can have a content. Toward what is thus proper to it, its relation is not at all formal. No science is completely formal; but even gravitational mechanics is formal to a certain degree, in so far as optical and chemical properties are all the same to it. There is some question, however, as to the extent to which Frege took logic to provide constitutive norms of thought.

Linnebo suggests that Frege eventually rejected this idea. But many Frege scholars are convinced that Frege took the laws of logic to provide constitutive norms of thought MacFarlane , Taschek , Steinberger MacFarlane, in particular, argues that Kant and Frege may have agreed that one of the most important characteristics of logic is its generality, and that this generality consists in the fact that it provides normative rules and prescriptions.

So, though they may differ as to which principles are logical, there may be at least one point of reconciliation concerning how Kant and Frege conceived of logic. After all, modern logicians and philosophers of logic have not yet come to agreement about the proper conception of logic. It is important to recognize just how much Frege took himself to be focusing on the content, as opposed to the form, of thoughts.

Dudman trans. Frege was at least as interested in formalizing the content of reasoning as he was in formulating the rules for deriving a given thought from some group of thoughts. Frege would not have regarded the logical axioms of his formal systems as axiom schemata, i.

Nor would he have agreed that the logical axioms of his system were uninterpreted sentences. We have seen that ambiguity simply has to be rejected … Instead, Frege thought that his logical axioms are a fundamental truths governing negation, conditionalization, quantification, identity, and description, and b principles from which other such fundamental truths could be derived.

Indeed, even though Frege sometimes introduces methods for abbreviating these truths, he takes great pains to insist that these abbreviations are to be understood in terms of the full content being expressed.

Blanchette nicely shows, both in the entry on the Frege-Hilbert controversy and in her book , Ch. The reader should pursue these works for a more detailed explanation and nuanced discussion of the disagreement. One can appreciate how Frege and Hilbert might have failed to engage with one another by considering a simple analogy.

So, for Frege, this would clearly be a case of logical consequence. This analogy might help one to see how Frege and Hilbert might differ in their approach to questions of consistency and interpretation. This issue is the subject of the first half of Blanchette To see what is at stake, we vary the example from the one used in Blanchette Her answer Chapter 4 is that the formal representation of the arithmetic law has to be self-evidently logically equivalent to a good analysis of the original.

The reader is directed to her work for discussion of this important point. His philosophy of language has had just as much, if not more, impact than his contributions to logic and mathematics.

In this paper, Frege considered two puzzles about language and noticed, in each case, that one cannot account for the meaningfulness or logical behavior of certain sentences simply on the basis of the denotations of the terms names and descriptions in the sentence. One puzzle concerned identity statements and the other concerned sentences with subordinate clauses such as propositional attitude reports.

To solve these puzzles, Frege suggested that the terms of a language have both a sense and a denotation, i. This idea has inspired research in the field for over a century and we discuss it in what follows. The morning star is identical to the evening star. Mark Twain is Samuel Clemens. In the latter cases, you have to do some arithmetical work or astronomical investigation to learn the truth of these identity claims.

A propositional attitude is a psychological relation between a person and a proposition. Belief, desire, intention, discovery, knowledge, etc.

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## Frege’s Theorem and Foundations for Arithmetic

Important Secondary Works 1. His full christened name was Friedrich Ludwig Gottlob Frege. Little is known about his youth. Both were also principals of the school at various points: Karl held the position until his death , when Auguste took over until her death in Frege probably lived in Wismar until ; in the years from he is known to have studied at the Gymnasium in Wismar.

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## Frege's theorem

Basic Law V also correctly implies the Principle of Extensionality. This principle asserts that if two extensions have the same members, they are identical. Frege quickly added an Appendix to the second volume, describing two distinct ways of deriving a contradiction from Basic Law V. He also suggested a way of repairing Law V, but Quine later showed that such a repair was disastrous, since it would force the domain of objects to contain at most one object. In the next subsections, we describe the two ways of deriving a contradiction from Basic Law V that Frege described in the Appendix to Gg.

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## Publications

In breve, Frege voleva dimostrare che, con la teoria degli insiemi e una buona definizione di logica, si potessero derivare tutti gli insiemi numerici. Riguardo a molti problemi particolari trovo nella sua opera discussioni, distinzioni e definizioni che si cercano invano nelle opere di altri logici. Specialmente per quel che riguarda le funzioni cap. Lei afferma p.

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## Gottlob Frege (1848—1925)

Indeed, for each condition defined above, the concepts that satisfy the condition are all pairwise equinumerous to one another. This extension contains all the concepts that satisfy Condition 0 above, and so the number of all such concepts is 0. Frege, however, had a deep idea about how to do this. Note that the last conjunct is true because there is exactly 1 object namely, Bertrand Russell that falls under the concept author of Principia Mathematica other than Whitehead.