# Tag Info

## Hot answers tagged mathematical-logic

19

This story bears characteristic signs of a tall tale, although in this case one can identify the origin. It appears to be an amalgamation of two anecdotes, neither of which is itself very credible. Both are traceable to Warren McCulloch, Pitts's co-author on "A Logical Calculus of Ideas Immanent in Nervous Activity" (1943), which proposed the first ...

18

The paper 'Some facts about Kurt Gödel' by Wang (1981) (regrettably paywalled) contains a section that suggests Hilbert was not present when Gödel originally announced his sketch of the First Incompleteness Theorem at Königsberg, on the 7th of September, 1930. Notable mathematicians that were present include Carnap, Heyting and most importantly von Neumann,...

16

Your guess is right. Russell's paradox broke only what people considered at that time as foundations of set theory. More specifically, the logical foundation system built by Frege. Of course this was very disturbing, because many people understood that logic and set theory is really the foundation of the whole mathematics. However it did not "invalidate" any ...

14

This question is based on a misunderstanding. The statement that $\pi$ is constant has precise meaning: $\pi$ is a ratio of the length of circumference to the length of diameter. The statement that it is constant means that it is the same for all circles. (This statement is independent of the representation of this ratio with digits). Contrary to what many ...

13

Constance Reid's biography Hilbert contains a brief discussion at the beginning of chapter 23. Hilbert was somewhat angry... but then began to try to deal constructively with the problem... Broadened methods would permit the loosening of the requirements of formalizing. Hilbert himself now took a step in this direction. This was the replacing of the ...

13

Nowadays, metamathematics is a standard part of the landscape of mathematical logic. On the one hand, most work on foundations of mathematics should probably be considered metamathematical. The standard foundation is set-theoretic, with ZFC and its variants being the usual formalizations. But this is by far not the only option and, for instance, there is ...

12

Russell was not the first to discover "his" paradox. By June 1901 when he arrived at it (it was not published until the first edition of Principia in 1903), it was already known for a while to the Hilbert's circle at Göttingen. Russell did not belong to that circle, so his rediscovery was independent, but he was familiar with Schröder's Algebra of ...

11

Vitali's construction is probably the first example "unpleasant" in the modern sense, but well-ordering of the continuum was unpleasant enough to some at the time of Zermelo's proof in 1904. Accepting that the continuum can be dissociated into points (contra Aristotle) was recent and hard enough, that it could be well-ordered strained credulity further. ...

10

Regarding logic, the answer is a mixed one. The development of medieval logic is wide area of study; see at least SEP's entries : Medieval Theories of the Syllogism Medieval Theories of Consequence Medieval Theories of Modality Medieval Theories: Properties of Terms and more ..., as well as : Dov Gabbay & John Woods (editors), Handbook of the ...

10

You can see : Giuseppe Peano , Lezioni di Analisi Infinitesimale, 2 vols., 1893, page 17 : $$[f(x)]_{x=a}=f(a).$$ Not sure it is the earliest... but Peano was a prolific "inventor of notations". Regarding : how they express "$y$ under the condition that $x=2$" see e.g. page 34 [shortened] : let $y$ the natural logarithm of $x$ [...] and $f(x) = \... 10 As Francois Ziegler notes, Kelley attributes it to Halmos. In the past (before Halmos), definitions might be given in the form A group is called abelian if$xy=yx$for all$x,y\$. ...and every mathematics student would need to be told "since this is a definition, if means if and only if." I was one of the students told this: in the 1960s many ...

10

It may be surprising, but the material implication does not come from truth tables, the truth table definition is a late development. Neither de Morgan, nor Peirce, nor Frege, nor even Russell came up with it or justified it by matching Boolean operations to something in Plato and Aristotle. A detailed story can be found in Cajori's History of mathematical ...

10

You might be looking for the Italian School of Algebraic Geometry. It has become the canonical example of problems with a lack of rigour. The short summary is that the school started with some unfounded postulates, that they used to derive a wide number of results. This must be understood in the context of an incipient field, where no rigorous foundation was ...

9

Let me open with a quote from Hofstadter's classic Godel-Escher-Bach:"In my opinion, if one is interested in understanding Gödel's proof in a deep way, then one must recognize that the proof, in its essence, consists of a fusion of these two main ideas. Each of them alone is a master stroke; to put them together took an act of genius. If I were to choose, ...

9

There are various more recent works on subjects that can be considered as metamathematics. For example, Reverse Mathematics was started by Harvey Friedman in the mid-seventies. Recently, there was quite a bit of excitement around Homotopy Type Theory and Univalent Foundations not only but also as it ties in nicely with effort to have automatically ...

9

Latin loan words in German usually retained their "c" until about the end of the 19th century. The modern spellings like Kontinuum (for Continuum) and Zentrum (for Centrum) result from a fairly recent spelling reform.

9

No, he was not. Cohen wrote his own account of the history of forcing, The Discovery of Forcing (Rocky Mountain J. Math. 32 (4) (2002), 1071-1100), where he addresses his relationship with Gödel in detail. In particular, he calls him "one of my heroes" (along with Skolem), and mentions some conversations "years later, after my own proof". Nonetheless, it ...

9

Mathematical logic is a quite modern discipline : it emerged in the mid-19th century with Boole, Peirce and Frege. Logic instead, is quite ancient : we can date it at least from Aristotle (384–322 BCE) and the Stoics. In ancient Greek mathematics we have poofs and axioms : see Euclid's Elements and Archimedes' treatises. Thus, the axiomatic method is at ...

9

When introducing the older terminology in the previous sentence, Peano describes it thus: ... signifie "il y a des a", "les a existent"... It seems likely this is the source of the inverted "E".

8

I've found in Hubert Kennedy, Twelve Articles on Giuseppe Peano (2002) : What Russell learned from Peano, from Notre Dame Journal of Formal Logic (1983) the following: The evidence comes from Russell himself in, among other places, the description of the International Congress of Philosophy in Paris, 1900, in his Autobiography (p.217–219): The ...

8

As I explained in my answer to your other question, mathematics was always done using ordinary (non-formalized) logic. Attempts to formalize logic begin with Aristotle. (This is called "formal logic"). At the later time, the idea arose to use mathematics to formalize logic. One of the early proponents of this idea was Leibniz, and it achieved further ...

8

Yes and no. It is better to say that there were always several different standards. Most proofs of Euclides, Archimedes and Apollonius are on the level of modern standards, though gaps in those proofs can be found, and some were found in the ancients time. At the time where calculus was invented, many mathematicians understood that the proofs using new ...

7

In principle, yes. See: Roman Murawski, Undefinability of Truth. The Problem of Priority: Tarski vs Gödel (1998): Tarski made clear his indebtedness to Gödel's methods. [...] On the other hand Tarski strongly emphasized the fact that his results were obtained independently. Tarski did not claim any priority for Gödel's own results. In footnote 2 to ...

7

Cantor did not invent the term, it goes back to antiquity. "Latin was a lingua franca, the learned language for scientific and political affairs, for more than a thousand years, being eventually replaced by French in the 18th century and English in the late 19th". In Latin, French and English continuum starts with "c". Cantor saw himself ...

7

Multiplication, before the invention of modern (axiomatic) algebra, was defined as the operation giving the area of a rectangle with sides of a particular length.1 Commutativity of multiplication then follows from two axioms: Congruent geometrical figures have equal areas Any geometrical figure reflected through a line is congruent to the original figure ...

7

I do not believe that degrees of computability were directly relevant to the topic of Gödel's 1946 lecture because they do not relativize computability in the relevant language dependent sense that Gödel stressed, and Parsons after him. Stillwell gives a lucid explanation of the difference in this regard between Turing computability (solvability, ...

7

In responding to your question, I hope that I can convince you of the following two claims: Gödel held that the formal system of the actual Principia was that of ramified type thoery (simple type theory with orders built into the syntax). Gödel held that the formal system of Principia without orders built into the syntax would have been motivated by a kind ...

6

The Tower of Hanoi – Myths and Maths gives a detailed historical account of the game, its predecessors and myths surrounding it, with references to original sources, it also goes in-depth into mathematical issues. The Temple of Brahma story was first reported in 1883 by Professor N. Claus (de Siam), mandarin of college Li-Sou-Stian, and was embellished by ...

6

I believe I know where the occasional lower case spelling (and thus your confusion) comes from. Take for example this book which contains "Selected papers from the 7th Augustus de Morgan Workshop, London". Note the lower case "de". The editors of the book were Dutch (Amsterdam University Press, Johan van Benthem); and in Dutch, words like "de" or "van" ...

6

This is one of those cases where it is easy to take modern ideas for granted, and wonder how something so "obvious" could be missed. Modern formal logic with its array of logical operations, Boolean algebra and predicates is the creation of late 19th century, only finalized after much labor by Russell and Whitehead around 1910. Logical analysis ...

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