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A lot of math symbols are accidental and owe their existence to circumstances of writing, book publishing, typewriter or later computer keyboard. Very few have

A lot of math symbols are accidental and owe their existence to circumstances of writing, book publishing, typewriter or later computer keyboard. Very few have any kind of history that helps understand their meaning. Just a few examples: "lambda" in "lambda calculus" was originally the caret sign, which, by convention was used on typewriters because those didn't have Greek letters. Whether Alonzo Church actually wanted the Greek letter or the caret symbol is not known (and he, himself, probably wouldn't care), but if you look at the manuscript, it uses "^". Minus sign was an attempt to create a sign "opposite to plus", whereas the plus sign was a contraction of Latin "at" (ampersand is another way in which Latin "at" made it into modern typesetting). Multiplication was also an attempt to manipulate the plus sign. And so was division (the horizontal line between two dots, and subsequent mutation into just two dots making a ":" sign). Asterisk, which used plenty in different mathematical contexts was actually a punctuation sign. Early Roman writing didn't use spaces between words, instead they put dots between them. It was the only punctuation sign they used, so no commas, periods, question marks etc. All that stuff came almost a thousand years later. A bunch of logical signs were invented by Frege and later popularized / amended by Wittgenstein. These were completely artificial and didn't derive from anything. I don't think the authors ever explained why they chose any specific shapes, but my guess would be that they were looking for something visually distinct enough for a small alphabet they created and arbitrarily assigned symbols to operations.

joe_the_user 3y ago
Interesting post but I think you're wrong in the following sentence: A bunch of logical signs were invented by Frege and later popularized / amended by Wittgenstein. Jan Von Plato goes into this in The Great Formal Machinery Works. Frege's notation never become popular - his notation involved lines from one part of an expression to another (!) and is fairly incomprehensible. A more or less modern system of symbols of mathematical logic was created by Russel and Whitehead in Principia Mathematica extending the symbols of Peano (but using the ideas of Frege). Russel and Whitehead syntax, ("there exists" and "for all" symbol) was taken up by Hilbert and the Gottingen school which crafted a lot of the basic theories of mathematical logic. Von Plato doesn't refer to Wittgenstein influencing notation and I don't think he did - he was a popular but idiosyncratic philosopher, not a mathematician. Von Plato's history of mathematical logic in Great Formal Machinery... is fascinating. Edit: Wikipedia describes Frege's notation. You have draw lines connecting expressions, an approach that quickly gets unwieldy. I believe that only Frege used Frege's notation. Wittgenstein refers to Frege but that's it. https://en.wikipedia.org/wiki/Begriffsschrift https://en.wikipedia.org/wiki/Begriffsschrift
- crabbone 3y ago
  To the best of my knowledge, Wittgenstein lived with Russel for quite some time and was a very influential force in writing the Principa. I believe I saw a lot of prototypes of the modern logical symbols in his Traktatus. I also believe that Russel traveled to meet Frege in his quest for writing the book. He definitely saw his work even before it was published / had private conversations with him face-to-face. Not sure how much of the invented language made it into Principia, but the author was definitely familiar with Frege's work.
  - joe_the_user 3y ago
    All I know is Von Plato and some random readings in foundations for my degree, but this[1] seems to indicate Russel met Wittgenstein after Principia was completed. I also believe that Russel traveled to meet Frege in his quest for writing the book. No doubt, as I said above, Principia was very influenced by Frege's ideas. But it absolutely did not use Frege's notation(which was a disaster as I describe above). Von Plato really emphasizes how much of an advance the notation of Principia was. And seriously, it's remarkable how new modern mathematical logic notation is, it didn't have true modern form 'till the 20s or 30s. [1] "It was October 1911, volume one of Principia Mathematica was newly out, and Bertrand Russell, fellow of Trinity College, Cambridge, was having his tea when a young man suddenly appeared. He introduced himself as "Loot'vig Vit'gun-shteyne". https://www.independent.co.uk/life-style/when-ludwig-wittgenstein-met-bertrand-russell-1596995.html https://www.independent.co.uk/life-style/when-ludwig-wittgen...
- codeflo 3y ago
  That Wikipedia section is uncharacteristically bad. It doesn’t even define or explain the notation, it’s just some examples and a long quote. I’m sure that there was more to it than that.
  - Hendrikto 3y ago
    The German version has more details and better explanations.
_0ffh 3y ago
Minor nitpick: "And" in latin is "et", as in "et al.".
- crabbone 3y ago
  Yes, whoops.
thaumasiotes 3y ago
> the plus sign was a contraction of Latin "at" (ampersand is another way in which Latin "at" made it into modern typesetting) An ampersand is et [and], not at [but]. There are still current forms of the symbol that are transparently just a T attached to an E. I was about to ask how the plus sign could possibly be related to the word "but", but I'm guessing you meant that it derives from et too?
User23 3y ago
> A bunch of logical signs were invented by Frege This is true. I find Peirce's existential graphs interesting because they permit stating existential and universal quantifiers without any explicit syntax! Frege's work evidently has priority by about 4 years, but it doesn't appear that Peirce was aware of it.

joe_the_user 3y ago

Interesting post but I think you're wrong in the following sentence: A bunch of logical signs were invented by Frege and later popularized / amended by Wittgenstein. Jan Von Plato goes into this in The Great Formal Machinery Works. Frege's notation never become popular - his notation involved lines from one part of an expression to another (!) and is fairly incomprehensible. A more or less modern system of symbols of mathematical logic was created by Russel and Whitehead in Principia Mathematica extending the symbols of Peano (but using the ideas of Frege). Russel and Whitehead syntax, ("there exists" and "for all" symbol) was taken up by Hilbert and the Gottingen school which crafted a lot of the basic theories of mathematical logic. Von Plato doesn't refer to Wittgenstein influencing notation and I don't think he did - he was a popular but idiosyncratic philosopher, not a mathematician. Von Plato's history of mathematical logic in Great Formal Machinery... is fascinating. Edit: Wikipedia describes Frege's notation. You have draw lines connecting expressions, an approach that quickly gets unwieldy. I believe that only Frege used Frege's notation. Wittgenstein refers to Frege but that's it. https://en.wikipedia.org/wiki/Begriffsschrift https://en.wikipedia.org/wiki/Begriffsschrift

crabbone 3y ago
To the best of my knowledge, Wittgenstein lived with Russel for quite some time and was a very influential force in writing the Principa. I believe I saw a lot of prototypes of the modern logical symbols in his Traktatus. I also believe that Russel traveled to meet Frege in his quest for writing the book. He definitely saw his work even before it was published / had private conversations with him face-to-face. Not sure how much of the invented language made it into Principia, but the author was definitely familiar with Frege's work.
- joe_the_user 3y ago
  All I know is Von Plato and some random readings in foundations for my degree, but this[1] seems to indicate Russel met Wittgenstein after Principia was completed. I also believe that Russel traveled to meet Frege in his quest for writing the book. No doubt, as I said above, Principia was very influenced by Frege's ideas. But it absolutely did not use Frege's notation(which was a disaster as I describe above). Von Plato really emphasizes how much of an advance the notation of Principia was. And seriously, it's remarkable how new modern mathematical logic notation is, it didn't have true modern form 'till the 20s or 30s. [1] "It was October 1911, volume one of Principia Mathematica was newly out, and Bertrand Russell, fellow of Trinity College, Cambridge, was having his tea when a young man suddenly appeared. He introduced himself as "Loot'vig Vit'gun-shteyne". https://www.independent.co.uk/life-style/when-ludwig-wittgenstein-met-bertrand-russell-1596995.html https://www.independent.co.uk/life-style/when-ludwig-wittgen...
codeflo 3y ago
That Wikipedia section is uncharacteristically bad. It doesn’t even define or explain the notation, it’s just some examples and a long quote. I’m sure that there was more to it than that.
- Hendrikto 3y ago
  The German version has more details and better explanations.

_0ffh 3y ago

Minor nitpick: "And" in latin is "et", as in "et al.".

crabbone 3y ago
Yes, whoops.

thaumasiotes 3y ago

> the plus sign was a contraction of Latin "at" (ampersand is another way in which Latin "at" made it into modern typesetting) An ampersand is et [and], not at [but]. There are still current forms of the symbol that are transparently just a T attached to an E. I was about to ask how the plus sign could possibly be related to the word "but", but I'm guessing you meant that it derives from et too?

User23 3y ago

> A bunch of logical signs were invented by Frege This is true. I find Peirce's existential graphs interesting because they permit stating existential and universal quantifiers without any explicit syntax! Frege's work evidently has priority by about 4 years, but it doesn't appear that Peirce was aware of it.