Is mathematics a syntax?
up vote
2
down vote
favorite
I have read that syntax is symbol and semantics is meaning those symbols convey.
- Is mathematics a syntax?
- Where is semantics in mathematics? What gives meaning to mathematics, to those symbols I write?
Probably any meaning Mathematics can ever have is the behaviour of its syntax only? But if syntax has a characteristic behaviour, is it a pure syntax after all? So it is a meaning oriented syntax which defines itself.
Or Mathematics is just an ideal syntax! If so, where is the semantics?
I can't figure out the interface of syntax and semantics pertaining to mathematics. Which part is syntax and which part is semantics? And what about logic?
logic philosophy
asked Nov 22 at 13:02
Ajax
966
|
show 4 more comments
up vote
2
down vote
favorite
I have read that syntax is symbol and semantics is meaning those symbols convey.
- Is mathematics a syntax?
- Where is semantics in mathematics? What gives meaning to mathematics, to those symbols I write?
Probably any meaning Mathematics can ever have is the behaviour of its syntax only? But if syntax has a characteristic behaviour, is it a pure syntax after all? So it is a meaning oriented syntax which defines itself.
Or Mathematics is just an ideal syntax! If so, where is the semantics?
I can't figure out the interface of syntax and semantics pertaining to mathematics. Which part is syntax and which part is semantics? And what about logic?
logic philosophy
asked Nov 22 at 13:02
Ajax
966
1
Arithmetic is the "theory of numbers". Formulas used in theorems are made according to the syntax. Formulas "speak of" numbers and their properties, which are the semantics of the theory.
– Mauro ALLEGRANZA
Nov 22 at 13:25
@MauroALLEGRANZA What constitutes the syntax? Is it a pure syntax?
– Ajax
Nov 22 at 13:40
The syntax of e.g. arithmetic is that of first-order arithmetic. In practice, we use many abbreviations.
– Mauro ALLEGRANZA
Nov 22 at 13:46
@MauroALLEGRANZA Are you saying that Peano Axioms constitute the syntax for some arithmetic?
– Ajax
Nov 22 at 13:50
Example : Ethan Bloch, The real numbers and real analysis, Springer (2011), page 3.
– Mauro ALLEGRANZA
Nov 22 at 14:13
|
show 4 more comments
up vote
2
down vote
favorite
up vote
2
down vote
favorite
I have read that syntax is symbol and semantics is meaning those symbols convey.
- Is mathematics a syntax?
- Where is semantics in mathematics? What gives meaning to mathematics, to those symbols I write?
Probably any meaning Mathematics can ever have is the behaviour of its syntax only? But if syntax has a characteristic behaviour, is it a pure syntax after all? So it is a meaning oriented syntax which defines itself.
Or Mathematics is just an ideal syntax! If so, where is the semantics?
I can't figure out the interface of syntax and semantics pertaining to mathematics. Which part is syntax and which part is semantics? And what about logic?
logic philosophy
asked Nov 22 at 13:02
Ajax
966
I have read that syntax is symbol and semantics is meaning those symbols convey.
- Is mathematics a syntax?
- Where is semantics in mathematics? What gives meaning to mathematics, to those symbols I write?
Probably any meaning Mathematics can ever have is the behaviour of its syntax only? But if syntax has a characteristic behaviour, is it a pure syntax after all? So it is a meaning oriented syntax which defines itself.
Or Mathematics is just an ideal syntax! If so, where is the semantics?
I can't figure out the interface of syntax and semantics pertaining to mathematics. Which part is syntax and which part is semantics? And what about logic?
logic philosophy
logic philosophy
asked Nov 22 at 13:02
Ajax
966
asked Nov 22 at 13:02
Ajax
966
edited Nov 22 at 13:08
asked Nov 22 at 13:02
Ajax
966
asked Nov 22 at 13:02
Ajax
966
asked Nov 22 at 13:02
Ajax
966
966
1
Arithmetic is the "theory of numbers". Formulas used in theorems are made according to the syntax. Formulas "speak of" numbers and their properties, which are the semantics of the theory.
– Mauro ALLEGRANZA
Nov 22 at 13:25
@MauroALLEGRANZA What constitutes the syntax? Is it a pure syntax?
– Ajax
Nov 22 at 13:40
The syntax of e.g. arithmetic is that of first-order arithmetic. In practice, we use many abbreviations.
– Mauro ALLEGRANZA
Nov 22 at 13:46
@MauroALLEGRANZA Are you saying that Peano Axioms constitute the syntax for some arithmetic?
– Ajax
Nov 22 at 13:50
Example : Ethan Bloch, The real numbers and real analysis, Springer (2011), page 3.
– Mauro ALLEGRANZA
Nov 22 at 14:13
|
show 4 more comments
1
Arithmetic is the "theory of numbers". Formulas used in theorems are made according to the syntax. Formulas "speak of" numbers and their properties, which are the semantics of the theory.
– Mauro ALLEGRANZA
Nov 22 at 13:25
@MauroALLEGRANZA What constitutes the syntax? Is it a pure syntax?
– Ajax
Nov 22 at 13:40
The syntax of e.g. arithmetic is that of first-order arithmetic. In practice, we use many abbreviations.
– Mauro ALLEGRANZA
Nov 22 at 13:46
@MauroALLEGRANZA Are you saying that Peano Axioms constitute the syntax for some arithmetic?
– Ajax
Nov 22 at 13:50
Example : Ethan Bloch, The real numbers and real analysis, Springer (2011), page 3.
– Mauro ALLEGRANZA
Nov 22 at 14:13
1
1
Arithmetic is the "theory of numbers". Formulas used in theorems are made according to the syntax. Formulas "speak of" numbers and their properties, which are the semantics of the theory.
– Mauro ALLEGRANZA
Nov 22 at 13:25
Arithmetic is the "theory of numbers". Formulas used in theorems are made according to the syntax. Formulas "speak of" numbers and their properties, which are the semantics of the theory.
– Mauro ALLEGRANZA
Nov 22 at 13:25
@MauroALLEGRANZA What constitutes the syntax? Is it a pure syntax?
– Ajax
Nov 22 at 13:40
@MauroALLEGRANZA What constitutes the syntax? Is it a pure syntax?
– Ajax
Nov 22 at 13:40
The syntax of e.g. arithmetic is that of first-order arithmetic. In practice, we use many abbreviations.
– Mauro ALLEGRANZA
Nov 22 at 13:46
The syntax of e.g. arithmetic is that of first-order arithmetic. In practice, we use many abbreviations.
– Mauro ALLEGRANZA
Nov 22 at 13:46
@MauroALLEGRANZA Are you saying that Peano Axioms constitute the syntax for some arithmetic?
– Ajax
Nov 22 at 13:50
@MauroALLEGRANZA Are you saying that Peano Axioms constitute the syntax for some arithmetic?
– Ajax
Nov 22 at 13:50
Example : Ethan Bloch, The real numbers and real analysis, Springer (2011), page 3.
– Mauro ALLEGRANZA
Nov 22 at 14:13
Example : Ethan Bloch, The real numbers and real analysis, Springer (2011), page 3.
– Mauro ALLEGRANZA
Nov 22 at 14:13
|
show 4 more comments
1 Answer
1
active
oldest
votes
up vote
1
down vote
I'm not a logician, a mathematician nor a philosopher but maybe I can try to (quite subjectively) answer to the question. It depends on what you call Mathematics and Syntax.
The separation between syntax and semantics isn't the only way to understand formal systems, if it works well for programming languages, it doesn't explain everything for Logic and Mathematics.
Is mathematics a syntax?
There's no objective answer but I would say that mathematics is first made of social conventions to convince other people with arguments we call proofs. To convey our thoughts we use written language thus syntax. Hovewer, syntax isn't necessary but sufficient to do mathematics. Between two humans the communication is done through language but we can also communicate with ourselves through our mind. Is any form of syntax is still involved? I don't know. To sum up, mathematics can be done through syntax but may exist without it.
The language of mathematics can indeed still be turned into a global syntax accepted by everyone but in the real world it is the interaction between local communities of mathematicians who make the existence of mathematics.
Where is semantics in mathematics? What gives meaning to mathematics, to those symbols I write?
I think it's quite clear that there's no semantics. Logic is different because we have a pre-conception of what logic is and what is right, wrong, true, false. The semantic is the meaning/format we want to give. But is there any meaning we want to give to mathematics? Unlike logic, mathematics is free of ideals (such as truth).
I think your intuition is right. Following Wittgenstein's philosophy, the meaning of mathematics may be only be determined by its use ("meaning is use" paradigm). Note that there're many communities in mathematics using symbols following their own conventions.
And what about logic?
I think that question is too broad. You may be interested in the Curry-Howard correspondence; a formal correspondance between computer programs and logical systems. Logical systems may be seen as emerging from coherent computational behaviours and interactions (no divergence). In mathematics and our everyday life, logical arguments are also what makes our discussion converge to a conclusion. Logical systems formally corresponds to type systems preventing infinite reductions. Semantical explanations work but doesn't explain why logic is as it is.
There're few works in that direction (Ludics, Geometry of Interaction, Transcendental Syntax, ...) mainly coming from Jean-Yves Girard.
answered Nov 27 at 10:57
Boris E.
388113
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3009102%2fis-mathematics-a-syntax%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
I'm not a logician, a mathematician nor a philosopher but maybe I can try to (quite subjectively) answer to the question. It depends on what you call Mathematics and Syntax.
The separation between syntax and semantics isn't the only way to understand formal systems, if it works well for programming languages, it doesn't explain everything for Logic and Mathematics.
Is mathematics a syntax?
There's no objective answer but I would say that mathematics is first made of social conventions to convince other people with arguments we call proofs. To convey our thoughts we use written language thus syntax. Hovewer, syntax isn't necessary but sufficient to do mathematics. Between two humans the communication is done through language but we can also communicate with ourselves through our mind. Is any form of syntax is still involved? I don't know. To sum up, mathematics can be done through syntax but may exist without it.
The language of mathematics can indeed still be turned into a global syntax accepted by everyone but in the real world it is the interaction between local communities of mathematicians who make the existence of mathematics.
Where is semantics in mathematics? What gives meaning to mathematics, to those symbols I write?
I think it's quite clear that there's no semantics. Logic is different because we have a pre-conception of what logic is and what is right, wrong, true, false. The semantic is the meaning/format we want to give. But is there any meaning we want to give to mathematics? Unlike logic, mathematics is free of ideals (such as truth).
I think your intuition is right. Following Wittgenstein's philosophy, the meaning of mathematics may be only be determined by its use ("meaning is use" paradigm). Note that there're many communities in mathematics using symbols following their own conventions.
And what about logic?
I think that question is too broad. You may be interested in the Curry-Howard correspondence; a formal correspondance between computer programs and logical systems. Logical systems may be seen as emerging from coherent computational behaviours and interactions (no divergence). In mathematics and our everyday life, logical arguments are also what makes our discussion converge to a conclusion. Logical systems formally corresponds to type systems preventing infinite reductions. Semantical explanations work but doesn't explain why logic is as it is.
There're few works in that direction (Ludics, Geometry of Interaction, Transcendental Syntax, ...) mainly coming from Jean-Yves Girard.
answered Nov 27 at 10:57
Boris E.
388113
add a comment |
up vote
1
down vote
I'm not a logician, a mathematician nor a philosopher but maybe I can try to (quite subjectively) answer to the question. It depends on what you call Mathematics and Syntax.
The separation between syntax and semantics isn't the only way to understand formal systems, if it works well for programming languages, it doesn't explain everything for Logic and Mathematics.
Is mathematics a syntax?
There's no objective answer but I would say that mathematics is first made of social conventions to convince other people with arguments we call proofs. To convey our thoughts we use written language thus syntax. Hovewer, syntax isn't necessary but sufficient to do mathematics. Between two humans the communication is done through language but we can also communicate with ourselves through our mind. Is any form of syntax is still involved? I don't know. To sum up, mathematics can be done through syntax but may exist without it.
The language of mathematics can indeed still be turned into a global syntax accepted by everyone but in the real world it is the interaction between local communities of mathematicians who make the existence of mathematics.
Where is semantics in mathematics? What gives meaning to mathematics, to those symbols I write?
I think it's quite clear that there's no semantics. Logic is different because we have a pre-conception of what logic is and what is right, wrong, true, false. The semantic is the meaning/format we want to give. But is there any meaning we want to give to mathematics? Unlike logic, mathematics is free of ideals (such as truth).
I think your intuition is right. Following Wittgenstein's philosophy, the meaning of mathematics may be only be determined by its use ("meaning is use" paradigm). Note that there're many communities in mathematics using symbols following their own conventions.
And what about logic?
I think that question is too broad. You may be interested in the Curry-Howard correspondence; a formal correspondance between computer programs and logical systems. Logical systems may be seen as emerging from coherent computational behaviours and interactions (no divergence). In mathematics and our everyday life, logical arguments are also what makes our discussion converge to a conclusion. Logical systems formally corresponds to type systems preventing infinite reductions. Semantical explanations work but doesn't explain why logic is as it is.
There're few works in that direction (Ludics, Geometry of Interaction, Transcendental Syntax, ...) mainly coming from Jean-Yves Girard.
answered Nov 27 at 10:57
Boris E.
388113
add a comment |
up vote
1
down vote
up vote
1
down vote
I'm not a logician, a mathematician nor a philosopher but maybe I can try to (quite subjectively) answer to the question. It depends on what you call Mathematics and Syntax.
The separation between syntax and semantics isn't the only way to understand formal systems, if it works well for programming languages, it doesn't explain everything for Logic and Mathematics.
Is mathematics a syntax?
There's no objective answer but I would say that mathematics is first made of social conventions to convince other people with arguments we call proofs. To convey our thoughts we use written language thus syntax. Hovewer, syntax isn't necessary but sufficient to do mathematics. Between two humans the communication is done through language but we can also communicate with ourselves through our mind. Is any form of syntax is still involved? I don't know. To sum up, mathematics can be done through syntax but may exist without it.
The language of mathematics can indeed still be turned into a global syntax accepted by everyone but in the real world it is the interaction between local communities of mathematicians who make the existence of mathematics.
Where is semantics in mathematics? What gives meaning to mathematics, to those symbols I write?
I think it's quite clear that there's no semantics. Logic is different because we have a pre-conception of what logic is and what is right, wrong, true, false. The semantic is the meaning/format we want to give. But is there any meaning we want to give to mathematics? Unlike logic, mathematics is free of ideals (such as truth).
I think your intuition is right. Following Wittgenstein's philosophy, the meaning of mathematics may be only be determined by its use ("meaning is use" paradigm). Note that there're many communities in mathematics using symbols following their own conventions.
And what about logic?
I think that question is too broad. You may be interested in the Curry-Howard correspondence; a formal correspondance between computer programs and logical systems. Logical systems may be seen as emerging from coherent computational behaviours and interactions (no divergence). In mathematics and our everyday life, logical arguments are also what makes our discussion converge to a conclusion. Logical systems formally corresponds to type systems preventing infinite reductions. Semantical explanations work but doesn't explain why logic is as it is.
There're few works in that direction (Ludics, Geometry of Interaction, Transcendental Syntax, ...) mainly coming from Jean-Yves Girard.
answered Nov 27 at 10:57
Boris E.
388113
I'm not a logician, a mathematician nor a philosopher but maybe I can try to (quite subjectively) answer to the question. It depends on what you call Mathematics and Syntax.
The separation between syntax and semantics isn't the only way to understand formal systems, if it works well for programming languages, it doesn't explain everything for Logic and Mathematics.
Is mathematics a syntax?
There's no objective answer but I would say that mathematics is first made of social conventions to convince other people with arguments we call proofs. To convey our thoughts we use written language thus syntax. Hovewer, syntax isn't necessary but sufficient to do mathematics. Between two humans the communication is done through language but we can also communicate with ourselves through our mind. Is any form of syntax is still involved? I don't know. To sum up, mathematics can be done through syntax but may exist without it.
The language of mathematics can indeed still be turned into a global syntax accepted by everyone but in the real world it is the interaction between local communities of mathematicians who make the existence of mathematics.
Where is semantics in mathematics? What gives meaning to mathematics, to those symbols I write?
I think it's quite clear that there's no semantics. Logic is different because we have a pre-conception of what logic is and what is right, wrong, true, false. The semantic is the meaning/format we want to give. But is there any meaning we want to give to mathematics? Unlike logic, mathematics is free of ideals (such as truth).
I think your intuition is right. Following Wittgenstein's philosophy, the meaning of mathematics may be only be determined by its use ("meaning is use" paradigm). Note that there're many communities in mathematics using symbols following their own conventions.
And what about logic?
I think that question is too broad. You may be interested in the Curry-Howard correspondence; a formal correspondance between computer programs and logical systems. Logical systems may be seen as emerging from coherent computational behaviours and interactions (no divergence). In mathematics and our everyday life, logical arguments are also what makes our discussion converge to a conclusion. Logical systems formally corresponds to type systems preventing infinite reductions. Semantical explanations work but doesn't explain why logic is as it is.
There're few works in that direction (Ludics, Geometry of Interaction, Transcendental Syntax, ...) mainly coming from Jean-Yves Girard.
answered Nov 27 at 10:57
Boris E.
388113
answered Nov 27 at 10:57
Boris E.
388113
answered Nov 27 at 10:57
Boris E.
388113
answered Nov 27 at 10:57
Boris E.
388113
388113
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3009102%2fis-mathematics-a-syntax%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
1
Arithmetic is the "theory of numbers". Formulas used in theorems are made according to the syntax. Formulas "speak of" numbers and their properties, which are the semantics of the theory.
– Mauro ALLEGRANZA
Nov 22 at 13:25
@MauroALLEGRANZA What constitutes the syntax? Is it a pure syntax?
– Ajax
Nov 22 at 13:40
The syntax of e.g. arithmetic is that of first-order arithmetic. In practice, we use many abbreviations.
– Mauro ALLEGRANZA
Nov 22 at 13:46
@MauroALLEGRANZA Are you saying that Peano Axioms constitute the syntax for some arithmetic?
– Ajax
Nov 22 at 13:50
Example : Ethan Bloch, The real numbers and real analysis, Springer (2011), page 3.
– Mauro ALLEGRANZA
Nov 22 at 14:13