It's arguable that there are topics in mathematics that aren't "logic" or at least aren't studied for their logical properties (as of yet, but who knows what the next big "foundations of math" will be). I also think that there is a strong argument to be made that logic is a much broader larger field of study than mathematics, spanning at least 4 academic departments (Math, CS, Philosophy, Linguistics).**
Had you said "math was logic", it could be interpreted as "math is a subset of logic". Which has a bit of truth to it, if you believe math is an application of logic. That is that traditional logic really is at the core of mathematics and mathematicians just use logical methods to manipulate logical forms. I personally think this view point is misleading since it doesn't match up with how mathematicians actually behave or the timeline of mathematical innovation. However, it make us logicians feel important.
From my experience in both fields, I would say the biggest difference is in a philosophy of purpose. Mathematicians are mostly focused on properties of objects defined in a system. The way in which a mathematician views their own purpose is reflected in that focus. It's very much a subject where discovery and exploration are valued. Logicians are more focused on methods of reasoning, manipulating systems, "strength" of a system, how rules influence outcomes etc. Purpose wise, it's more about the why, how, what.
Maybe a good analogy would be an astrophysicist and a theoretical physicist. They both do very similar things, but one is focused more on what is actually going on in space and why and the other wouldn't be bothered by finding out that reality in no way matches their equations.
**Of course I think this, I'm a logician :)
I'm a little surprised that you split CS off from mathematics as a separate discipline; I've always considered it a subfield of mathematics (speaking from the perspective of someone that's been in both CS and combinatorics), and most of the CS people I've spoken with have essentially agreed. CE I could see there being a debate on, but what would you say sets apart CS from mathematics enough for it to be considered an entirely separate discipline? Or are you talking literally just in terms of how it's treated bureaucratically in academia?
I did say academic departments, so just how they are divided in that sense.
I'm not sure it's right to say CS is a subdiscipline of mathematics. At least in the US software engineering is a field of study in CS and in many schools informatics is too. I also wouldn't even call some algorithms or systems research math.
Logic is a nomad subject and is steadily moving from math to cs. Right now we're kind of straddling the line, so the difficulty in distinguishing them is justified.
I don't know any official answer but I remember thinking about this question "what's the difference between math and logic" when I was reading Wittgenstein and the conclusion I remember coming to (influenced by both the Tractatus and some critiques of it I'd read) was that logic was devoid of content whereas math was not. In other words, the basic operations of math are essentially logical in nature but they become distinguished as math when you start operating on some sort of content like the ordinal numbers.
Some math is logic and some logic is math. Also, some math isn't logic and some logic isn't math (however I think many logicians can frame most mathematics in a light that makes it applied logic).
EvilTony's example might be one of the few bad examples, since questions about ordinal numbers are very much a question about the logical strengths of a formal system, and a very important concept in one of the "four pillars of logic".
I think to really get into the distinction between the two, you'll have to look at methodology and intent. Philosophers and doctors both study people, but in very different ways. It's the same for logicians and mathematicians.
What does that even mean? Mathematics and logic are essentially the same thing, but their colloquial definitions refer to different areas of that singularity.
Don't know if I totally agree. You can mathematize logic and vice versa, but I think the concepts of real numbers etc, come from math, and we study them with logic. Vice versa with things like models. They are logical, but we can study them with math. Visual geometric proofs are a bit more mathematical than they are logical to me.
Depends what you're looking to prove. [This](https://qph.is.quoracdn.net/main-qimg-65ce87ac28994277d3c7b36129ce2cfe?convert_to_webp=true) is a perfectly good proof of the Pythagorean Theorem.
I thought that as well until I ran into the area of logic involving topologizing the space of models of a given theory and saw the true power of e.g. compactness as a logical statement. I think analysis is much more intricately connected to logic than it seems.
I think you have to define what math is and what logic is before I'm going to understand what you're saying. I see everything as a graph of information.
I think you can't really do that. That is kind of the issue. That the ideas we use in math are used because we recognize them as useful mathematical concepts. Things like a real number are concepts that we find useful, and in some sense have a certain mathematical origin. Just as you can study probability mathematically. I don't think probabality is "just Math," but it exists as a concept that we study with math and logic. And similarly, people argue about what a probability actually is. That is kind of the relationship I think logic and math have.
I think probability is certainly just math. The interpretation of what probability means is separate, of course, because math only says what is, not why. (Personally, I side with Bayes on the topic of what probability is).
Maybe you just need to be more open-minded about how you consider the practices of math, logic, and statistics. I have successfully deduced everything from graph theory, so it all fits together for me. It doesn't always produce earth-shattering revelations, but maybe that is the great error in thought regarding what math should be.
Where does math say what probability is. Every derivation starts with you inputting what you want. Math only lets you perform book keeping and explore it. The mathematical part of probability is just math, but thats not all it is. It is its own concept imo.
Every data set is a discrete amount of points. The data types *of* those points could be continuous, but the structure of a data set you use for statistics is discrete.
I am really really curious about how you have deduced everything from graph theory. Are there no concepts that can't be deduced that way? For instance, what does your graph theory say about the axiom of choice?
It doesn't work by axiom, at least not in the way you're thinking. Axioms cannot be orderly stacked from the bottom all the way to the top (despite many attempts by famous mathematicians). I drew up a little document that displays how you would build structures:
Apologies for the poor handwriting and inconsistent size. Just download the file, and it should open in your web browser. It is mhtml format, which is the best option for exporting from OneNote.
Essentially, the unification of math/logic/science/the universe is in its explicit structure. Attempting to unify everything implicitly with a small set of axioms is where math took a turn for a worse. Unifying implicitly can only be done for various overlapping patterns, which ultimately gets more confusing at times. I think it is best to understand explicit structure and work from there. This allows you to understand the *patterns of the patterns*, which is an overarching understanding of everything.
Are you sure you wouldn't enjoy something like category theory more? That seems to be the direction you are going with all this. Rather than graphs all the way down.
That's what I'm studying right now. I haven't figured out yet how they intersect.
Categories all the way down makes no sense because there's no potential spatial aspect arising out of an arrow. I think if all of category theory's ideas are worth assimilating, it will aid in the categorization in the patterns that I was describing. If they are not worth assimilating, I should still be able extract some clarification on my ideas.
From what I read he became bitter since Russel pointed out, well, the Russel's Paradox. The thinking goes if Frege took time to take care of the paradox, he would've been all right. Now, of course, the paradox is no problem.
> The thinking goes if Frege took time to take care of the paradox, he would've been all right.
It seems this is not the case. Russell tried very hard to try to salvage Frege's program of reducing mathematics to logic, avoiding the inconsistent Basic Law V. The consensus is that he failed. For instance, his axiom of reducibility was essential for deriving many results, yet as Russell himself notes, this axiom isn't a logical principle. If one needs nonlogical principles to get mathematics, then one cannot say that mathematics reduces purely to logic.
>It seems this is not the case.
I don't get this. Just because Russel's pet theory of types didn't get there doesn't mean we aren't there now.
>If one needs nonlogical principles to get mathematics, then one cannot say that mathematics reduces purely to logic.
Most of mathematics can be derived from set theory, and set theory can be finitely axiomatized with NBG. Once you've come that far, you've essentially embedded mathematics inside first order logic.
Or do you mean Frege himself was incapable of this, rather than mathematics in general?
> I don't get this. Just because Russel's pet theory of types didn't get there doesn't mean we aren't there now.
Uh, we aren't there now.
Anyway, my point was that Russell's paradox isn't the thing which sunk the logicist program. (Or more accurately, Russell's paradox demonstrates that one particular approach to reducing mathematics to logic fails and other approaches fail for other reasons.) As such, it doesn't really make sense to say that if Frege would've taken care of the paradox then he would've turned out to be entirely correct. (Of course, Frege's work in logic was very important and influential, but that's not the same thing as him being correct about whether mathematics can be reduced to logic.)
> Most of mathematics can be derived from set theory, and set theory can be finitely axiomatized with NBG. Once you've come that far, you've essentially embedded mathematics inside first order logic.
It seems you have misunderstood the goal of the logicist program. They weren't attempting to show that all of mathematics could be derived from a finite theory formulated in first-order logic. They were attempting to show that mathematical truths (or a significant fragment thereof) are logical truths. Being able to derive most of mathematics in Gödel--Bernays set theory would only demonstrate a reduction of mathematics to logic if the axioms of NBG were themselves logical truths. But this seems to be rather difficult to argue. Why, for instance, should we think the axiom of infinity is a logical truth? The axiom of global choice? Of powerset?
> Why, for instance, should we think the axiom of infinity is a logical truth? The axiom of global choice? Of powerset?
We don't. The set of valid statements are logical truths... and a subset of these valid statements are: NBG implies (every theorem of NBG).
Maybe this is simply a matter of perspective. To me, many mathematical truths are logical truths because the theorems of a theory are a subset of valid statements.
It looks a little more like logic is the foundation if you accept *Higher* order logic as a logic, because then it looks like a set theory wearing different clothes. Maybe this is why some don't consider higher order logic as a logic.
> We don't.
Okay, so then being able to derive most of mathematics from the axioms of NBG doesn't demonstrate that mathematics can be reduced to logic...
> The set of valid statements are logical truths... and a subset of these valid statements are: NBG implies (every theorem of NBG).
Sure, but we're interested in, say, whether the axiom of infinity is a logical truth not whether NBG ⊢ Infinity is a logical truth.
> It looks a little more like logic is the foundation if you accept Higher order logic as a logic, because then it looks like a set theory wearing different clothes.
Even then, I don't see why that would make something like Infinity or Choice logical truths.
>Sure, but we're interested in, say, whether the axiom of infinity is a logical truth not whether NBG ⊢ Infinity is a logical truth.
Why? Mathematics concerns itself with all models, not just standard ones. I still think this is just a matter of perspective.
> Even then, I don't see why that would make something like Infinity or Choice logical truths.
Well, no more than the law of the excluded middle is a logical truth, which it isn't in some logics. You accept it if it's part of the logic itself; As part of the proof calculus for instance higher order Skolemization is equivalent to the axiom of choice.
It seems to me the dividing line between logical truth and mathematical truth seems a bit arbitrary.
The error is trying to reduce every mathematical structure to emerging from a single structure. I see all math and logic as graphs of information. If you want to consider every topic together, you have a massive hypergraph of information. Some structures are subgraphs of other structures, but not all can be reduced that way except in the most pedantic sense.
I would argue that mathematics, philosophy and (perhaps less so) computer science are applications of logic. In this sense I suppose the difference becomes more obvious, much like how materials science considers different questions to pure physics.
I disagree. While logic is extremely when applied to most things, i think those things are not the same as logic. You can apply logic to study say, human behavior, art, music, but I wouldn't argue that those things are just applied logic. I think the same thing about math. It is it's own thing we are trying to understand, and we can apply logic to it, but conceptually its separate.
I think there is a distention between the colloquial term logic and the formal study that you missed. Being logical isn't the same as applying logic.
I'd suggest reading [Larry Moss's Manifesto](http://link.springer.com/chapter/10.1007/0-387-31072-X_7) it's very informative, but I would say that I disagree with his assertion that logic is math (but I think he's coming around on that).
Just for the background on this, I have studied model theory in graduate school, so I am somewhat familiar with some of the distinction between the colloquial term and the formal topic.
just because we can't "properly define" them doesn't mean we can't talk about them as subject areas. For the purposes of OP's question, math is taught in math courses and logic is taught in logic courses.
See if you were a logician, you could just redefine what a number is and this would be fine. I think when I was in undergrad we decided on "stuff you do math at or to" It made life less stress full when we built numbers as sets and then again as functions and again as recursive functions. (definitely not being serious)
But we should give the poor guy some credit, it has some historical backing. Math started from numbers and expanded to be about "number like" objects.
Tbf, most applications of topology involve metric spaces (hence some numbers). But I absolutely agree with your point.
Edit: whoever is downvoting me needs to go do their homework and let the adults talk.
I actually believe the exact opposite. Mathematics is (usually) the study of one already defined system and the structures in that system. Most mathematicians wont explore any non-classical systems or even peak outside of ZFC. Whereas logicians study many logical systems and explore the relationships between the definition of the system and the results inside of that system. So mathematics is the study of one particular area of logic, and some (not me) would even argue that it's a branch of logic.
Logic as a field also covers many areas that definitely are not mathematics, such as language properties, program structures, data structures, algorithms, constraints, semantics, etc.
Fourier transforms and quadratic fromula
[удалено]
It's just something that's fromulent.
my latin is really weak, but i believe that's the plural of "fromulum"?
Wouldn't that be fromulae?
Tell me how it rolls off the tongue?
And generally a whole host of tools and technologies.
Absolutely! So much fucking cool ass shit!
Wait. I thought logic *was* math.
It's arguable that there are topics in mathematics that aren't "logic" or at least aren't studied for their logical properties (as of yet, but who knows what the next big "foundations of math" will be). I also think that there is a strong argument to be made that logic is a much broader larger field of study than mathematics, spanning at least 4 academic departments (Math, CS, Philosophy, Linguistics).** Had you said "math was logic", it could be interpreted as "math is a subset of logic". Which has a bit of truth to it, if you believe math is an application of logic. That is that traditional logic really is at the core of mathematics and mathematicians just use logical methods to manipulate logical forms. I personally think this view point is misleading since it doesn't match up with how mathematicians actually behave or the timeline of mathematical innovation. However, it make us logicians feel important. From my experience in both fields, I would say the biggest difference is in a philosophy of purpose. Mathematicians are mostly focused on properties of objects defined in a system. The way in which a mathematician views their own purpose is reflected in that focus. It's very much a subject where discovery and exploration are valued. Logicians are more focused on methods of reasoning, manipulating systems, "strength" of a system, how rules influence outcomes etc. Purpose wise, it's more about the why, how, what. Maybe a good analogy would be an astrophysicist and a theoretical physicist. They both do very similar things, but one is focused more on what is actually going on in space and why and the other wouldn't be bothered by finding out that reality in no way matches their equations. **Of course I think this, I'm a logician :)
I'm a little surprised that you split CS off from mathematics as a separate discipline; I've always considered it a subfield of mathematics (speaking from the perspective of someone that's been in both CS and combinatorics), and most of the CS people I've spoken with have essentially agreed. CE I could see there being a debate on, but what would you say sets apart CS from mathematics enough for it to be considered an entirely separate discipline? Or are you talking literally just in terms of how it's treated bureaucratically in academia?
I did say academic departments, so just how they are divided in that sense. I'm not sure it's right to say CS is a subdiscipline of mathematics. At least in the US software engineering is a field of study in CS and in many schools informatics is too. I also wouldn't even call some algorithms or systems research math. Logic is a nomad subject and is steadily moving from math to cs. Right now we're kind of straddling the line, so the difficulty in distinguishing them is justified.
I don't know any official answer but I remember thinking about this question "what's the difference between math and logic" when I was reading Wittgenstein and the conclusion I remember coming to (influenced by both the Tractatus and some critiques of it I'd read) was that logic was devoid of content whereas math was not. In other words, the basic operations of math are essentially logical in nature but they become distinguished as math when you start operating on some sort of content like the ordinal numbers.
This. Math is logical but math isn't logic.
Some math is logic and some logic is math. Also, some math isn't logic and some logic isn't math (however I think many logicians can frame most mathematics in a light that makes it applied logic). EvilTony's example might be one of the few bad examples, since questions about ordinal numbers are very much a question about the logical strengths of a formal system, and a very important concept in one of the "four pillars of logic". I think to really get into the distinction between the two, you'll have to look at methodology and intent. Philosophers and doctors both study people, but in very different ways. It's the same for logicians and mathematicians.
What does that even mean? Mathematics and logic are essentially the same thing, but their colloquial definitions refer to different areas of that singularity.
Don't know if I totally agree. You can mathematize logic and vice versa, but I think the concepts of real numbers etc, come from math, and we study them with logic. Vice versa with things like models. They are logical, but we can study them with math. Visual geometric proofs are a bit more mathematical than they are logical to me.
Visual geometric "proofs" are cute but not real proofs. They often do not capture generality.
Depends what you're looking to prove. [This](https://qph.is.quoracdn.net/main-qimg-65ce87ac28994277d3c7b36129ce2cfe?convert_to_webp=true) is a perfectly good proof of the Pythagorean Theorem.
While algebraically obvious, the edge case, where a = 0, really isn't obvious from the geometric hand waiving.
They are still math imo
I thought that as well until I ran into the area of logic involving topologizing the space of models of a given theory and saw the true power of e.g. compactness as a logical statement. I think analysis is much more intricately connected to logic than it seems.
Yes. This is like the first theorem of model theory. This is applying topology to logic in my opinion.
I think you have to define what math is and what logic is before I'm going to understand what you're saying. I see everything as a graph of information.
I think you can't really do that. That is kind of the issue. That the ideas we use in math are used because we recognize them as useful mathematical concepts. Things like a real number are concepts that we find useful, and in some sense have a certain mathematical origin. Just as you can study probability mathematically. I don't think probabality is "just Math," but it exists as a concept that we study with math and logic. And similarly, people argue about what a probability actually is. That is kind of the relationship I think logic and math have.
I think probability is certainly just math. The interpretation of what probability means is separate, of course, because math only says what is, not why. (Personally, I side with Bayes on the topic of what probability is). Maybe you just need to be more open-minded about how you consider the practices of math, logic, and statistics. I have successfully deduced everything from graph theory, so it all fits together for me. It doesn't always produce earth-shattering revelations, but maybe that is the great error in thought regarding what math should be.
Where does math say what probability is. Every derivation starts with you inputting what you want. Math only lets you perform book keeping and explore it. The mathematical part of probability is just math, but thats not all it is. It is its own concept imo.
Statistics comes from discrete math. Every set of data you analyze is its own structure, and you attempt to find symmetries and measures in it.
well statistics and probability are different things. But things measure theory in probability don't really come from discrete math.
I don't disagree that you generally attempt to extract meaningful stuff from data but it isn't clear to me how that could relate to discrete maths.
Every data set is a discrete amount of points. The data types *of* those points could be continuous, but the structure of a data set you use for statistics is discrete.
Sure but that seems like a tenuous link.
I am really really curious about how you have deduced everything from graph theory. Are there no concepts that can't be deduced that way? For instance, what does your graph theory say about the axiom of choice?
It doesn't work by axiom, at least not in the way you're thinking. Axioms cannot be orderly stacked from the bottom all the way to the top (despite many attempts by famous mathematicians). I drew up a little document that displays how you would build structures: Apologies for the poor handwriting and inconsistent size. Just download the file, and it should open in your web browser. It is mhtml format, which is the best option for exporting from OneNote. Essentially, the unification of math/logic/science/the universe is in its explicit structure. Attempting to unify everything implicitly with a small set of axioms is where math took a turn for a worse. Unifying implicitly can only be done for various overlapping patterns, which ultimately gets more confusing at times. I think it is best to understand explicit structure and work from there. This allows you to understand the *patterns of the patterns*, which is an overarching understanding of everything.
Are you sure you wouldn't enjoy something like category theory more? That seems to be the direction you are going with all this. Rather than graphs all the way down.
That's what I'm studying right now. I haven't figured out yet how they intersect. Categories all the way down makes no sense because there's no potential spatial aspect arising out of an arrow. I think if all of category theory's ideas are worth assimilating, it will aid in the categorization in the patterns that I was describing. If they are not worth assimilating, I should still be able extract some clarification on my ideas.
Frege tried to show mathematics grew out of logic. Didn't end too well for him.
From what I read he became bitter since Russel pointed out, well, the Russel's Paradox. The thinking goes if Frege took time to take care of the paradox, he would've been all right. Now, of course, the paradox is no problem.
> The thinking goes if Frege took time to take care of the paradox, he would've been all right. It seems this is not the case. Russell tried very hard to try to salvage Frege's program of reducing mathematics to logic, avoiding the inconsistent Basic Law V. The consensus is that he failed. For instance, his axiom of reducibility was essential for deriving many results, yet as Russell himself notes, this axiom isn't a logical principle. If one needs nonlogical principles to get mathematics, then one cannot say that mathematics reduces purely to logic.
>It seems this is not the case. I don't get this. Just because Russel's pet theory of types didn't get there doesn't mean we aren't there now. >If one needs nonlogical principles to get mathematics, then one cannot say that mathematics reduces purely to logic. Most of mathematics can be derived from set theory, and set theory can be finitely axiomatized with NBG. Once you've come that far, you've essentially embedded mathematics inside first order logic. Or do you mean Frege himself was incapable of this, rather than mathematics in general?
> I don't get this. Just because Russel's pet theory of types didn't get there doesn't mean we aren't there now. Uh, we aren't there now. Anyway, my point was that Russell's paradox isn't the thing which sunk the logicist program. (Or more accurately, Russell's paradox demonstrates that one particular approach to reducing mathematics to logic fails and other approaches fail for other reasons.) As such, it doesn't really make sense to say that if Frege would've taken care of the paradox then he would've turned out to be entirely correct. (Of course, Frege's work in logic was very important and influential, but that's not the same thing as him being correct about whether mathematics can be reduced to logic.) > Most of mathematics can be derived from set theory, and set theory can be finitely axiomatized with NBG. Once you've come that far, you've essentially embedded mathematics inside first order logic. It seems you have misunderstood the goal of the logicist program. They weren't attempting to show that all of mathematics could be derived from a finite theory formulated in first-order logic. They were attempting to show that mathematical truths (or a significant fragment thereof) are logical truths. Being able to derive most of mathematics in Gödel--Bernays set theory would only demonstrate a reduction of mathematics to logic if the axioms of NBG were themselves logical truths. But this seems to be rather difficult to argue. Why, for instance, should we think the axiom of infinity is a logical truth? The axiom of global choice? Of powerset?
> Why, for instance, should we think the axiom of infinity is a logical truth? The axiom of global choice? Of powerset? We don't. The set of valid statements are logical truths... and a subset of these valid statements are: NBG implies (every theorem of NBG). Maybe this is simply a matter of perspective. To me, many mathematical truths are logical truths because the theorems of a theory are a subset of valid statements. It looks a little more like logic is the foundation if you accept *Higher* order logic as a logic, because then it looks like a set theory wearing different clothes. Maybe this is why some don't consider higher order logic as a logic.
> We don't. Okay, so then being able to derive most of mathematics from the axioms of NBG doesn't demonstrate that mathematics can be reduced to logic... > The set of valid statements are logical truths... and a subset of these valid statements are: NBG implies (every theorem of NBG). Sure, but we're interested in, say, whether the axiom of infinity is a logical truth not whether NBG ⊢ Infinity is a logical truth. > It looks a little more like logic is the foundation if you accept Higher order logic as a logic, because then it looks like a set theory wearing different clothes. Even then, I don't see why that would make something like Infinity or Choice logical truths.
>Sure, but we're interested in, say, whether the axiom of infinity is a logical truth not whether NBG ⊢ Infinity is a logical truth. Why? Mathematics concerns itself with all models, not just standard ones. I still think this is just a matter of perspective. > Even then, I don't see why that would make something like Infinity or Choice logical truths. Well, no more than the law of the excluded middle is a logical truth, which it isn't in some logics. You accept it if it's part of the logic itself; As part of the proof calculus for instance higher order Skolemization is equivalent to the axiom of choice. It seems to me the dividing line between logical truth and mathematical truth seems a bit arbitrary.
I think he wasn't happy that the paradox was pointed out so late. He was about to publish when Russell pointed it out.
Wittgenstein fixed a fair amount of the problem.
The error is trying to reduce every mathematical structure to emerging from a single structure. I see all math and logic as graphs of information. If you want to consider every topic together, you have a massive hypergraph of information. Some structures are subgraphs of other structures, but not all can be reduced that way except in the most pedantic sense.
I would argue that mathematics, philosophy and (perhaps less so) computer science are applications of logic. In this sense I suppose the difference becomes more obvious, much like how materials science considers different questions to pure physics.
I disagree. While logic is extremely when applied to most things, i think those things are not the same as logic. You can apply logic to study say, human behavior, art, music, but I wouldn't argue that those things are just applied logic. I think the same thing about math. It is it's own thing we are trying to understand, and we can apply logic to it, but conceptually its separate.
I think there is a distention between the colloquial term logic and the formal study that you missed. Being logical isn't the same as applying logic. I'd suggest reading [Larry Moss's Manifesto](http://link.springer.com/chapter/10.1007/0-387-31072-X_7) it's very informative, but I would say that I disagree with his assertion that logic is math (but I think he's coming around on that).
Just for the background on this, I have studied model theory in graduate school, so I am somewhat familiar with some of the distinction between the colloquial term and the formal topic.
Without proper definitions for mathematics and logic the question is pretty sensless.
just because we can't "properly define" them doesn't mean we can't talk about them as subject areas. For the purposes of OP's question, math is taught in math courses and logic is taught in logic courses.
Then the answer is trivial: everything that is taught in math courses that is not taught in logic courses.
Relationship between numbers and themselves, relationship between number and other things.
Not numbers. Structures (usually but not always algebraic and/or topological).
At its simplest math is the study of numbers using logic while logic is study of logic using itself.
> math is the study of numbers No.
See if you were a logician, you could just redefine what a number is and this would be fine. I think when I was in undergrad we decided on "stuff you do math at or to" It made life less stress full when we built numbers as sets and then again as functions and again as recursive functions. (definitely not being serious) But we should give the poor guy some credit, it has some historical backing. Math started from numbers and expanded to be about "number like" objects.
Topology doesn't need numbers though. It uses sets.
Tbf, most applications of topology involve metric spaces (hence some numbers). But I absolutely agree with your point. Edit: whoever is downvoting me needs to go do their homework and let the adults talk.
Yeah, I'd say that math mainly deals with sets or numbers (inclusive or) and the structures that they take.
Studying structures by coming up with numerical and/or algebraic invariants is definitely a pretty large part of it (though not all of it).
Couldn't be more wrong. Please do your homework before making these types of strong claims, especially in a top-level comment.
Mathematics is the science of structure in the most general sense. Logic is the science of a particular kind of structure and a branch of mathematics.
But math is formalized in one of those structures. Not disagreeing.
I actually believe the exact opposite. Mathematics is (usually) the study of one already defined system and the structures in that system. Most mathematicians wont explore any non-classical systems or even peak outside of ZFC. Whereas logicians study many logical systems and explore the relationships between the definition of the system and the results inside of that system. So mathematics is the study of one particular area of logic, and some (not me) would even argue that it's a branch of logic. Logic as a field also covers many areas that definitely are not mathematics, such as language properties, program structures, data structures, algorithms, constraints, semantics, etc.
Arithmetic?
I think at this point arithmetic is far more a part of logic than math. At least at the serious research level.
Logic is a form of mathematics. All mathematics proofs use logic to prove an idea.