8 ms·

The Mathematics of Charles Sanders Peirce (2001) [pdf]

55 points by language 10y ago

dmfdmf 10y ago
Not sure why this is trending on HN (and without any comments or discussion). TBH, its reads like a Sokal Hoax and seems like gibberish to me. https://en.wikipedia.org/wiki/Sokal_affair https://en.wikipedia.org/wiki/Sokal_affair
- gballan 10y ago
  Peirce's diagramatic logic is the forerunner to Sowa's conceptual graphs [0]. Worth a look. [0] https://en.m.wikipedia.org/wiki/Conceptual_graph https://en.m.wikipedia.org/wiki/Conceptual_graph
- MrQuincle 10y ago
  Not sure either. Semiotics is a philosophical branch of linguistics. De Saussure and Peirce I put conceptually in the same tradition of Cyc and other symbolic/ontological/semantic efforts in knowledge engineering. That being said, symbol grounding does not get enough attention in current connectionist systems. How to properly present grammar, time, pointers, signs?
- ykler 10y ago
  This is not like the Sokal hoax at all. Sure the writer has a few English departmentish affectations, but the article is basically just explaining Peirce's logic. And Peirce is one of roughly two people in history to figure out quantification.
S_Daedalus 10y ago
Like using a set of fine chisels to sculpt a pudding... math for semiotics. Pft.
- dang 10y ago
  Please stop posting unsubstantive comments to Hacker News.
  - S_Daedalus 10y ago
    While I have you here, is it acceptable to call people pedants here? https://news.ycombinator.com/item?id=12401447 https://news.ycombinator.com/item?id=12401447
    - dang 10y ago
      No.
      - S_Daedalus 10y ago
        Thank you.
        dang 10y ago
        You rewrote your comment in such a deceptive way that I've banned your account. I've also killed this pointless subthread.
pavlov 10y ago
In a comments subthread that was flag-killed, gballan made the following interesting link: "Peirce's diagramatic logic is the forerunner to Sowa's conceptual graphs [0]. Worth a look. [0] https://en.m.wikipedia.org/wiki/Conceptual_graph https://en.m.wikipedia.org/wiki/Conceptual_graph
todd8 10y ago
I couldn't get through all of the paper. It seemed to be simply a new, not especially useful, notation for certain boolean operators. The concept of universal boolean operators, in the sense of functional completeness (see [1]) isn't that deep. A binary boolean operator (function) has four possible combinations of inputs: TT, TF, FT, and FF. For example, the and operation is defined by (T and T) == T, (T and F) == F, (F and T) == F, and (F and F) = F. The familiar truth table [2] is a clearer way to write this out: A | B | A and B =============== T T T T F F F T F F F F Other boolean operators can be defined by truth tables too. How many unique boolean operators are there? Since there are four possible binary outputs there are 16 possible boolean operators. They are mostly very familiar, but mathematicians use various notations for them. Again, truth tables make them obvious: A B | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ==================================================== T T T T T T T T T T F F F F F F F F T F T T T T F F F F T T T T F F F F F T T T F F T T F F T T F F T T F F F F T F T F T F T F T F T F T F T F the 16 possible operators are thus: 1: true, sometimes indicated by the "top" character in math, it looks like a capital T 2: or 3: if, (A if B) means B implies A, written <== 4: A, just the value of the first argument 5: onlyif, (A onlyif B) is sometimes written ==> 6: B, the value of the second argument 7: iff, same as == 8: and 9: nand, not and 10: xor 11: !B, the negation of second argument 12: notonlyif, i.e. not(A ==> B) 13: !A, the negation of the first argument 14: notif, i.e. not(A <== B) 15: nor, not or 16: false, sometimes written in math with the bottom symbol which looks like an upside down capital T We don't normally use all of these operators in predicate logic or programming or digital circuits because a few familiar ones will do. In fact, if we allow ourselves the negation operator (i.e. not or ! in some programming languages) then all binary boolean operations can be obtain from only and and not or from or and not. For example the operator numbered 5 above is the implication operator (A ==> B), but this is simply (not A or B). The sign of illation in the original article is simply operation 5 above, A ==> B. Interestingly, two of the binary boolean operators are universal or functionally complete by themselves, nand and nor. Any stateless boolean circuit can be constructed out of only nand gates or only nor gates. A programming language could get by with only one boolean operator, the binary nand (or the binary nor). For example ((not A) and B) could be rewritten using only nand: ((not A) and B) == (A nand A) and B == not ( (A nand A) nand B ) == ((A nand A) nand B) nand ((A nand A) nand B) [1] https://en.wikipedia.org/wiki/Functional_completeness https://en.wikipedia.org/wiki/Functional_completeness [2] https://en.wikipedia.org/wiki/Truth_table https://en.wikipedia.org/wiki/Truth_table
abecedarius 10y ago
Peirce also remarked in a letter how electrical relay circuits correspond one-for-one with boolean logic formulas. He seems to have been the first to notice this, almost 50 years before anyone else. His writings are very tedious to me, though, and I hope this article I just glanced through makes him more palatable.
mrcactu5 10y ago
similar: Lou Kauffman's "Box Arithmetic" http://homepages.math.uic.edu/~kauffman/Arithmetic.htm http://homepages.math.uic.edu/~kauffman/Arithmetic.htm

55 points by language 10y ago

Not sure why this is trending on HN (and without any comments or discussion). TBH, its reads like a Sokal Hoax and seems like gibberish to me. https://en.wikipedia.org/wiki/Sokal_affair https://en.wikipedia.org/wiki/Sokal_affair

gballan 10y ago
Peirce's diagramatic logic is the forerunner to Sowa's conceptual graphs [0]. Worth a look. [0] https://en.m.wikipedia.org/wiki/Conceptual_graph https://en.m.wikipedia.org/wiki/Conceptual_graph
MrQuincle 10y ago
Not sure either. Semiotics is a philosophical branch of linguistics. De Saussure and Peirce I put conceptually in the same tradition of Cyc and other symbolic/ontological/semantic efforts in knowledge engineering. That being said, symbol grounding does not get enough attention in current connectionist systems. How to properly present grammar, time, pointers, signs?
ykler 10y ago
This is not like the Sokal hoax at all. Sure the writer has a few English departmentish affectations, but the article is basically just explaining Peirce's logic. And Peirce is one of roughly two people in history to figure out quantification.

S_Daedalus 10y ago

Like using a set of fine chisels to sculpt a pudding... math for semiotics. Pft.

dang 10y ago
Please stop posting unsubstantive comments to Hacker News.
- S_Daedalus 10y ago
  While I have you here, is it acceptable to call people pedants here? https://news.ycombinator.com/item?id=12401447 https://news.ycombinator.com/item?id=12401447
  - dang 10y ago
    No.
    - S_Daedalus 10y ago
      Thank you.
      - dang 10y ago
        You rewrote your comment in such a deceptive way that I've banned your account. I've also killed this pointless subthread.

pavlov 10y ago

In a comments subthread that was flag-killed, gballan made the following interesting link: "Peirce's diagramatic logic is the forerunner to Sowa's conceptual graphs [0]. Worth a look. [0] https://en.m.wikipedia.org/wiki/Conceptual_graph https://en.m.wikipedia.org/wiki/Conceptual_graph

todd8 10y ago

I couldn't get through all of the paper. It seemed to be simply a new, not especially useful, notation for certain boolean operators. The concept of universal boolean operators, in the sense of functional completeness (see [1]) isn't that deep. A binary boolean operator (function) has four possible combinations of inputs: TT, TF, FT, and FF. For example, the and operation is defined by (T and T) == T, (T and F) == F, (F and T) == F, and (F and F) = F. The familiar truth table [2] is a clearer way to write this out: A | B | A and B =============== T T T T F F F T F F F F Other boolean operators can be defined by truth tables too. How many unique boolean operators are there? Since there are four possible binary outputs there are 16 possible boolean operators. They are mostly very familiar, but mathematicians use various notations for them. Again, truth tables make them obvious: A B | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ==================================================== T T T T T T T T T T F F F F F F F F T F T T T T F F F F T T T T F F F F F T T T F F T T F F T T F F T T F F F F T F T F T F T F T F T F T F T F the 16 possible operators are thus: 1: true, sometimes indicated by the "top" character in math, it looks like a capital T 2: or 3: if, (A if B) means B implies A, written <== 4: A, just the value of the first argument 5: onlyif, (A onlyif B) is sometimes written ==> 6: B, the value of the second argument 7: iff, same as == 8: and 9: nand, not and 10: xor 11: !B, the negation of second argument 12: notonlyif, i.e. not(A ==> B) 13: !A, the negation of the first argument 14: notif, i.e. not(A <== B) 15: nor, not or 16: false, sometimes written in math with the bottom symbol which looks like an upside down capital T We don't normally use all of these operators in predicate logic or programming or digital circuits because a few familiar ones will do. In fact, if we allow ourselves the negation operator (i.e. not or ! in some programming languages) then all binary boolean operations can be obtain from only and and not or from or and not. For example the operator numbered 5 above is the implication operator (A ==> B), but this is simply (not A or B). The sign of illation in the original article is simply operation 5 above, A ==> B. Interestingly, two of the binary boolean operators are universal or functionally complete by themselves, nand and nor. Any stateless boolean circuit can be constructed out of only nand gates or only nor gates. A programming language could get by with only one boolean operator, the binary nand (or the binary nor). For example ((not A) and B) could be rewritten using only nand: ((not A) and B) == (A nand A) and B == not ( (A nand A) nand B ) == ((A nand A) nand B) nand ((A nand A) nand B) [1] https://en.wikipedia.org/wiki/Functional_completeness https://en.wikipedia.org/wiki/Functional_completeness [2] https://en.wikipedia.org/wiki/Truth_table https://en.wikipedia.org/wiki/Truth_table

abecedarius 10y ago

Peirce also remarked in a letter how electrical relay circuits correspond one-for-one with boolean logic formulas. He seems to have been the first to notice this, almost 50 years before anyone else. His writings are very tedious to me, though, and I hope this article I just glanced through makes him more palatable.

mrcactu5 10y ago

similar: Lou Kauffman's "Box Arithmetic" http://homepages.math.uic.edu/~kauffman/Arithmetic.htm http://homepages.math.uic.edu/~kauffman/Arithmetic.htm