Mathematical realism and anti-realism

[𝕭𝖆𝖕𝖙𝖎𝖘𝖙𝖆 𝕸𝖎𝖓𝖔𝖑𝖆]

I realized I have never seen discussion of this on this board, and I would like to know what everyone thinks. Possibly the central question in philosophy of Mathematics: do mathematical objects exist (and/or have a truth value) independently of the human mind?

Personally I am inclined to answer yes, and in a way that is fairly metaphysical or even spiritual.

[Hnv1]

What are mathematical objects here? A euclidean construction of a square on a piece of paper? numbers? sets? axioms of propositional logic p->(q->p)? homotopy relation?

That's a pretty big question. There is a sense where I think intuitionism is correct and there are some mathematical properties of the world, there is some sense in which I have two socks on me right now without appealing to any abstract notion of a number and "I am wearing two socks" is true. What does this truthfulness bring about though? is it mind-independent? not sure

However, I lean to conventionalism about mathematics and in general about meanings. Any mathematical property of the world can be described by other equivalent or similar mathematical formalization and there is no one true mathematical system of reality. Abstract mathematics like 7-dimensional geometry is true in an empty tautological sense.

But bottom line, this is not the hill I'll die on. I can live with mathematical strict realism whereas I can't with metaethical realism.

[palandio]

I would say yes.

If nature exists outside the human mind, then there is that kind of structure in it that is as mathematical as life is biological and matter and force are physical. Now there may be different mathematical formalizations that exist only in our minds, but there is an underlying reality. And going even further I don't see why there would be reality behind 2- and 3-dimensional geometry, but not behind higher-dimensional geometry. I mean 6-dimensional geometry is used to describe the position of physical objects like a tool (5-dimensional if the tool is rotating), if we want to describe the phase space we're already at 12 dimensions, 11 if energy is conserved. And then there probably is a reality/truth behind n-dimensional geometry in general. And behind topology, and so on.

But there are reasons why I'm not a philosopher. I'm wilfully ignorant when it comes to philosophy of Mathematics, it makes my head ache. I'm just a simple man trained in a mathematical discipline that is usually considered Applied Math although it has increasingly moved into the abstract direction during the last decades. If someone put a gun to my head and asked if Mathematics were a Naturwissenschaft or a Geisteswissenschaft, I would say Naturwissenschaft, unlike many pure Mathematicians among whom it has become fashionable to consider Mathematics a Geisteswissenschaft, but they don't know what a Geisteswissenschaft really is, I think.

(If this sounded too sarcastic for you, I actually enjoy discussing with philosophers a lot, but I also enjoy just doing and applying math without constantly reflecting about the philosophical essence of what I'm doing.)

[Okay, maybe Mike Johnson is a competent parliamentarian.]

I think it's pretty obvious except in the absolute most nose-bleedingly high Idealist metaphysical systems that mathematical concepts describe aspects of the mind-independent physical world. Is the question here about "objects", i.e. whether, for example, the process of division is a "thing" as opposed to an abstract class imposed on the world by the human need to understand it? That strikes me as a fairly typical case of the problem of universals rather than anything special to do with mathematics, unless there's something I'm missing here.

[Hnv1]

I would say yes.

If nature exists outside the human mind, then there is that kind of structure in it that is as mathematical as life is biological and matter and force are physical. Now there may be different mathematical formalizations that exist only in our minds, but there is an underlying reality. And going even further I don't see why there would be reality behind 2- and 3-dimensional geometry, but not behind higher-dimensional geometry. I mean 6-dimensional geometry is used to describe the position of physical objects like a tool (5-dimensional if the tool is rotating), if we want to describe the phase space we're already at 12 dimensions, 11 if energy is conserved. And then there probably is a reality/truth behind n-dimensional geometry in general. And behind topology, and so on.

But there are reasons why I'm not a philosopher. I'm wilfully ignorant when it comes to philosophy of Mathematics, it makes my head ache. I'm just a simple man trained in a mathematical discipline that is usually considered Applied Math although it has increasingly moved into the abstract direction during the last decades. If someone put a gun to my head and asked if Mathematics were a Naturwissenschaft or a Geisteswissenschaft, I would say Naturwissenschaft, unlike many pure Mathematicians among whom it has become fashionable to consider Mathematics a Geisteswissenschaft, but they don't know what a Geisteswissenschaft really is, I think.

(If this sounded too sarcastic for you, I actually enjoy discussing with philosophers a lot, but I also enjoy just doing and applying math without constantly reflecting about the philosophical essence of what I'm doing.)

That's an interesting position and it sounds similar to Paul Benacerraf's structuralism in mathematics and perhaps entity realism in sciences.

It's that platonic theories like numbers exist as some abstract ideas hovering above the ultimate reality that usually gives ire to anti-realists like me, not the softer realism.

I find it odd when someone says the number 2 as an object must exist in order for us to use it in math. It seems unnecessary for me as there is nothing fundamental about numbers. There is a sense in which the number two appears in:
there is an (x), there is a (y) such that: P applies to x & P applies to y & x=/=y
without being an object

[Statilius the Epicurean]

I think it's pretty obvious except in the absolute most nose-bleedingly high Idealist metaphysical systems that mathematical concepts describe aspects of the mind-independent physical world. Is the question here about "objects", i.e. whether, for example, the process of division is a "thing" as opposed to an abstract class imposed on the world by the human need to understand it? That strikes me as a fairly typical case of the problem of universals rather than anything special to do with mathematics, unless there's something I'm missing here.

Numbers/mathematics are generally considered abstract objects, which is a different category to universals. If one thinks of a red apple, it has the property of redness but it doesn't have the property of the number 1 (or anything mathematical). The number 1 is an abstract representation of any singular object in a mathematical formula. There have been many people who hold to realism/Platonism about abstract objects like numbers and are nominalists in the case of universals, such as Quine. And vice-versa.

Anyway, I don't really have a position on mathematical Platonism other than a minimalist metpahysical temperament which inclines me to be suspicious of claims like things such as numbers exist separate from our brains. But I haven't really read about the issue. I'm more familiar with the problem of universals too.

[Hnv1]

I think it's pretty obvious except in the absolute most nose-bleedingly high Idealist metaphysical systems that mathematical concepts describe aspects of the mind-independent physical world. Is the question here about "objects", i.e. whether, for example, the process of division is a "thing" as opposed to an abstract class imposed on the world by the human need to understand it? That strikes me as a fairly typical case of the problem of universals rather than anything special to do with mathematics, unless there's something I'm missing here.

Numbers/mathematics are generally considered abstract objects, which is a different category to universals. If one thinks of a red apple, it has the property of redness but it doesn't have the property of the number 1 (or anything mathematical). The number 1 is an abstract representation of any singular object in a mathematical formula. There have been many people who hold to realism/Platonism about abstract objects like numbers and are nominalists in the case of universals, such as Quine. And vice-versa.

Anyway, I don't really have a position on mathematical Platonism other than a minimalist metpahysical temperament which inclines me to be suspicious of claims like things such as numbers exist separate from our brains. But I haven't really read about the issue. I'm more familiar with the problem of universals too.

I'm not sure Quine can be regarded full realist about numbers. He argued that objects exist for so long as we need them for our best scientific theory (indispensability argument). That's a pragmatic criterion, if we develop a better replacement in the future to numbers they can be cast aside. Let's not forget that for most of the early science quantification played no part and physics preferred dealing with ordinal measurements over cardinals.

Indeed universals is a different problem. But I think that you have a problem with mathematical objects you would probably have a bigger problem with "Redness" or "Motherness" just floating about. Though realists there usually appeal to "the joints of nature" which they can't do for numbers

[palandio]

I would say yes.

If nature exists outside the human mind, then there is that kind of structure in it that is as mathematical as life is biological and matter and force are physical. Now there may be different mathematical formalizations that exist only in our minds, but there is an underlying reality. And going even further I don't see why there would be reality behind 2- and 3-dimensional geometry, but not behind higher-dimensional geometry. I mean 6-dimensional geometry is used to describe the position of physical objects like a tool (5-dimensional if the tool is rotating), if we want to describe the phase space we're already at 12 dimensions, 11 if energy is conserved. And then there probably is a reality/truth behind n-dimensional geometry in general. And behind topology, and so on.

But there are reasons why I'm not a philosopher. I'm wilfully ignorant when it comes to philosophy of Mathematics, it makes my head ache. I'm just a simple man trained in a mathematical discipline that is usually considered Applied Math although it has increasingly moved into the abstract direction during the last decades. If someone put a gun to my head and asked if Mathematics were a Naturwissenschaft or a Geisteswissenschaft, I would say Naturwissenschaft, unlike many pure Mathematicians among whom it has become fashionable to consider Mathematics a Geisteswissenschaft, but they don't know what a Geisteswissenschaft really is, I think.

(If this sounded too sarcastic for you, I actually enjoy discussing with philosophers a lot, but I also enjoy just doing and applying math without constantly reflecting about the philosophical essence of what I'm doing.)

That's an interesting position and it sounds similar to Paul Benacerraf's structuralism in mathematics and perhaps entity realism in sciences.

It's that platonic theories like numbers exist as some abstract ideas hovering above the ultimate reality that usually gives ire to anti-realists like me, not the softer realism.

I find it odd when someone says the number 2 as an object must exist in order for us to use it in math. It seems unnecessary for me as there is nothing fundamental about numbers. There is a sense in which the number two appears in:
there is an (x), there is a (y) such that: P applies to x & P applies to y & x=/=y
without being an object

Thank you for the suggestions. I will try to read more about Benaceraff, entity realism, structural realism, etc.

Platonic realism and generally the more extreme forms of realism seem a bit naive to me. But still I would find it discomforting to consider mathematics to exist just in our minds, just because we need to use abstraction. If you think about it, 'life', 'cell', 'organism', 'animal', etc. are abstractions (or universals?) as much as '2' or 'space' are. And scientific realism in the natural sciences outside mathematics is still dominant, although maybe a higher dose of healthy skepticism would be adequate.

The irony is that in the field in which I have the most experience, i.e. probability, I'm agnostic about the reality of random itself. Not only can probability theory be put on a purely axiomatic foundation, but it can be applied in any context where you quantify lack of information (the deFinettian approach). At the same time I like to think that there is e.g. a real structure outside the human mind to which the Central Limit Theorem (and the various versions of it) gives us an insight.

One last thing about objects. Years ago a senior student asked one of my geometry professors: "Excuse me, what is geometry according to you? It's all about objects, isn't it?" And the professor said: "No, no, no. It's about spaces with structures on them and their relations." And this in my opinion applies to most of mathematics. Mathematical objects alone are almost meaningless constructions of our mind. It is their interactions that hopefully can give us a glimpse on interactions that are going on outside our minds.

[Statilius the Epicurean]

I'm not sure Quine can be regarded full realist about numbers. He argued that objects exist for so long as we need them for our best scientific theory (indispensability argument). That's a pragmatic criterion, if we develop a better replacement in the future to numbers they can be cast aside. Let's not forget that for most of the early science quantification played no part and physics preferred dealing with ordinal measurements over cardinals.

Quine was a realist about abstract objects because that's what the indispensability criterion requires: we must believe mathematical objects are real in order for our best scientific theories to work, and Quine believed philosophy was continuous with science and we should subordinate our metaphysical views to our best scientific theories. If one takes that to be a pragmatic belief one might say that our belief in gravity or atoms or plate tectonics is only pragmatic. But I don't think Quine would say he believed in gravity or atoms or plate tectonics or abstract objects merely because they are useful, but also because our best scientific theories give us reason to believe they are true descriptions of reality.

Indeed universals is a different problem. But I think that you have a problem with mathematical objects you would probably have a bigger problem with "Redness" or "Motherness" just floating about. Though realists there usually appeal to "the joints of nature" which they can't do for numbers

I agree that nominalists about mathematical objects will generally be inclined to be nominalists about universals due to broadly similar concerns between the two positions. Just was saying it's not a necessary connection.

[𝕭𝖆𝖕𝖙𝖎𝖘𝖙𝖆 𝕸𝖎𝖓𝖔𝖑𝖆]

I find it interesting that the responses so far seem to be skeptical of Platonism, as I would think it is the closest to my views. To elaborate on the OP, I tend to believe mathematical objects and structures are real and exist a priori, but aren't really empirical or concrete in the same way as what we call "physical objects", although they describe so many aspects of physical reality (possibly they are 'embedded' in it - but I am not too sure what this word fully entails). On a related note, I quite dislike to consider Mathematics a natural science, if perhaps a science at all, which I hope will not appear inconsistent.

As I alluded to, this intersects with my spirituality to some extent, which surely adds to the chasm. I do feel like this sort of connection may be underappreciated, especially compared to say how religious beliefs obviously influence metaethical views. But I have said enough about me already.

[Hnv1]

I find it interesting that the responses so far seem to be skeptical of Platonism, as I would think it is the closest to my views. To elaborate on the OP, I tend to believe mathematical objects and structures are real and exist a priori, but aren't really empirical or concrete in the same way as what we call "physical objects", although they describe so many aspects of physical reality (possibly they are 'embedded' in it - but I am not too sure what this word fully entails). On a related note, I quite dislike to consider Mathematics a natural science, if perhaps a science at all, which I hope will not appear inconsistent.

As I alluded to, this intersects with my spirituality to some extent, which surely adds to the chasm. I do feel like this sort of connection may be underappreciated, especially compared to say how religious beliefs obviously influence metaethical views. But I have said enough about me already.

I don't know about a priori, this is usually an epistemic category (if we follow Kripke here). Perhaps mind-independent.

The problem here is that sciences themselves use a variety of natural kinds that can be formalized in nature but are different theoretically. Take biology, some of the models in computational biology can indeed be deemed the description of the thing in itself. But then take something like clades, we can do all sort of mathematical models of them but we will be reluctant to say clades and their properties exist without our taxonomy of species into clades.

[Hnv1]

I'm not sure Quine can be regarded full realist about numbers. He argued that objects exist for so long as we need them for our best scientific theory (indispensability argument). That's a pragmatic criterion, if we develop a better replacement in the future to numbers they can be cast aside. Let's not forget that for most of the early science quantification played no part and physics preferred dealing with ordinal measurements over cardinals.

Quine was a realist about abstract objects because that's what the indispensability criterion requires: we must believe mathematical objects are real in order for our best scientific theories to work, and Quine believed philosophy was continuous with science and we should subordinate our metaphysical views to our best scientific theories. If one takes that to be a pragmatic belief one might say that our belief in gravity or atoms or plate tectonics is only pragmatic. But I don't think Quine would say he believed in gravity or atoms or plate tectonics or abstract objects merely because they are useful, but also because our best scientific theories give us reason to believe they are true descriptions of reality.

Indeed universals is a different problem. But I think that you have a problem with mathematical objects you would probably have a bigger problem with "Redness" or "Motherness" just floating about. Though realists there usually appeal to "the joints of nature" which they can't do for numbers

I agree that nominalists about mathematical objects will generally be inclined to be nominalists about universals due to broadly similar concerns between the two positions. Just was saying it's not a necessary connection.

As I read Quine I think his position is what we should believe, not what is really out there. I know his naturalized epistemology tried to do away with normativity in belief but I read the indispensability argument saying we are justified and ought to believe what our best scientific theories currently require. But if we conjoin this lesson with the notorious holism of Two Dogmas there is no belief that is in principle "analytic" or "a priori" that it cannot in principle be revised in the future. I take this whole project to full short of full-blooded realism.

Lewis, who infamously was a realist about both sets and "possible worlds", actually in his 1987 made a similar argument. We should be realists about possible worlds because its beneficial and we need it (for our semantic theories), but that falls short of full-blooded realism as well with the pragmatic wedge in between.

I personally don't have a horse in this race, being Sellarsian I accept that there is never any "given" to begin with.

[Hnv1]

I would say yes.

If nature exists outside the human mind, then there is that kind of structure in it that is as mathematical as life is biological and matter and force are physical. Now there may be different mathematical formalizations that exist only in our minds, but there is an underlying reality. And going even further I don't see why there would be reality behind 2- and 3-dimensional geometry, but not behind higher-dimensional geometry. I mean 6-dimensional geometry is used to describe the position of physical objects like a tool (5-dimensional if the tool is rotating), if we want to describe the phase space we're already at 12 dimensions, 11 if energy is conserved. And then there probably is a reality/truth behind n-dimensional geometry in general. And behind topology, and so on.

But there are reasons why I'm not a philosopher. I'm wilfully ignorant when it comes to philosophy of Mathematics, it makes my head ache. I'm just a simple man trained in a mathematical discipline that is usually considered Applied Math although it has increasingly moved into the abstract direction during the last decades. If someone put a gun to my head and asked if Mathematics were a Naturwissenschaft or a Geisteswissenschaft, I would say Naturwissenschaft, unlike many pure Mathematicians among whom it has become fashionable to consider Mathematics a Geisteswissenschaft, but they don't know what a Geisteswissenschaft really is, I think.

(If this sounded too sarcastic for you, I actually enjoy discussing with philosophers a lot, but I also enjoy just doing and applying math without constantly reflecting about the philosophical essence of what I'm doing.)

That's an interesting position and it sounds similar to Paul Benacerraf's structuralism in mathematics and perhaps entity realism in sciences.

It's that platonic theories like numbers exist as some abstract ideas hovering above the ultimate reality that usually gives ire to anti-realists like me, not the softer realism.

I find it odd when someone says the number 2 as an object must exist in order for us to use it in math. It seems unnecessary for me as there is nothing fundamental about numbers. There is a sense in which the number two appears in:
there is an (x), there is a (y) such that: P applies to x & P applies to y & x=/=y
without being an object

Thank you for the suggestions. I will try to read more about Benaceraff, entity realism, structural realism, etc.

Platonic realism and generally the more extreme forms of realism seem a bit naive to me. But still I would find it discomforting to consider mathematics to exist just in our minds, just because we need to use abstraction. If you think about it, 'life', 'cell', 'organism', 'animal', etc. are abstractions (or universals?) as much as '2' or 'space' are. And scientific realism in the natural sciences outside mathematics is still dominant, although maybe a higher dose of healthy skepticism would be adequate.

The irony is that in the field in which I have the most experience, i.e. probability, I'm agnostic about the reality of random itself. Not only can probability theory be put on a purely axiomatic foundation, but it can be applied in any context where you quantify lack of information (the deFinettian approach). At the same time I like to think that there is e.g. a real structure outside the human mind to which the Central Limit Theorem (and the various versions of it) gives us an insight.

One last thing about objects. Years ago a senior student asked one of my geometry professors: "Excuse me, what is geometry according to you? It's all about objects, isn't it?" And the professor said: "No, no, no. It's about spaces with structures on them and their relations." And this in my opinion applies to most of mathematics. Mathematical objects alone are almost meaningless constructions of our mind. It is their interactions that hopefully can give us a glimpse on interactions that are going on outside our minds.

Do you as a mathematician feel intuitionist logic and category theory to be "realer" and better foundation to mathematics than standard logic and set theory? Category theory at least on paper tries to ground relation in spatial terms

[Statilius the Epicurean]

I'm not sure Quine can be regarded full realist about numbers. He argued that objects exist for so long as we need them for our best scientific theory (indispensability argument). That's a pragmatic criterion, if we develop a better replacement in the future to numbers they can be cast aside. Let's not forget that for most of the early science quantification played no part and physics preferred dealing with ordinal measurements over cardinals.

Quine was a realist about abstract objects because that's what the indispensability criterion requires: we must believe mathematical objects are real in order for our best scientific theories to work, and Quine believed philosophy was continuous with science and we should subordinate our metaphysical views to our best scientific theories. If one takes that to be a pragmatic belief one might say that our belief in gravity or atoms or plate tectonics is only pragmatic. But I don't think Quine would say he believed in gravity or atoms or plate tectonics or abstract objects merely because they are useful, but also because our best scientific theories give us reason to believe they are true descriptions of reality.

Indeed universals is a different problem. But I think that you have a problem with mathematical objects you would probably have a bigger problem with "Redness" or "Motherness" just floating about. Though realists there usually appeal to "the joints of nature" which they can't do for numbers

I agree that nominalists about mathematical objects will generally be inclined to be nominalists about universals due to broadly similar concerns between the two positions. Just was saying it's not a necessary connection.

As I read Quine I think his position is what we should believe, not what is really out there. I know his naturalized epistemology tried to do away with normativity in belief but I read the indispensability argument saying we are justified and ought to believe what our best scientific theories currently require. But if we conjoin this lesson with the notorious holism of Two Dogmas there is no belief that is in principle "analytic" or "a priori" that it cannot in principle be revised in the future. I take this whole project to full short of full-blooded realism.

Lewis, who infamously was a realist about both sets and "possible worlds", actually in his 1987 made a similar argument. We should be realists about possible worlds because its beneficial and we need it (for our semantic theories), but that falls short of full-blooded realism as well with the pragmatic wedge in between.

I personally don't have a horse in this race, being Sellarsian I accept that there is never any "given" to begin with.

Apparently Quine explicitly addressed scientific realism in "Posits and Reality"

Having noted that man has no evidence for the existence of bodies beyond the fact that their assumption helps him organize experience, we should have done well, instead of disclaiming evidence for the existence of bodies, to conclude: such then, at bottom, is what evidence is, both for ordinary bodies and for molecules. . . .

So for Quine, our belief in the best scientific theories is continuous with our own sensory data, in the manner of his scientific holism. We have no reason to pragmatically believe in molecules or abstract objects any more than we should only pragmatically believe in the chair we're sitting in.

[Hnv1]

I'm not sure Quine can be regarded full realist about numbers. He argued that objects exist for so long as we need them for our best scientific theory (indispensability argument). That's a pragmatic criterion, if we develop a better replacement in the future to numbers they can be cast aside. Let's not forget that for most of the early science quantification played no part and physics preferred dealing with ordinal measurements over cardinals.

Quine was a realist about abstract objects because that's what the indispensability criterion requires: we must believe mathematical objects are real in order for our best scientific theories to work, and Quine believed philosophy was continuous with science and we should subordinate our metaphysical views to our best scientific theories. If one takes that to be a pragmatic belief one might say that our belief in gravity or atoms or plate tectonics is only pragmatic. But I don't think Quine would say he believed in gravity or atoms or plate tectonics or abstract objects merely because they are useful, but also because our best scientific theories give us reason to believe they are true descriptions of reality.

Indeed universals is a different problem. But I think that you have a problem with mathematical objects you would probably have a bigger problem with "Redness" or "Motherness" just floating about. Though realists there usually appeal to "the joints of nature" which they can't do for numbers

I agree that nominalists about mathematical objects will generally be inclined to be nominalists about universals due to broadly similar concerns between the two positions. Just was saying it's not a necessary connection.

As I read Quine I think his position is what we should believe, not what is really out there. I know his naturalized epistemology tried to do away with normativity in belief but I read the indispensability argument saying we are justified and ought to believe what our best scientific theories currently require. But if we conjoin this lesson with the notorious holism of Two Dogmas there is no belief that is in principle "analytic" or "a priori" that it cannot in principle be revised in the future. I take this whole project to full short of full-blooded realism.

Lewis, who infamously was a realist about both sets and "possible worlds", actually in his 1987 made a similar argument. We should be realists about possible worlds because its beneficial and we need it (for our semantic theories), but that falls short of full-blooded realism as well with the pragmatic wedge in between.

I personally don't have a horse in this race, being Sellarsian I accept that there is never any "given" to begin with.

Apparently Quine explicitly addressed scientific realism in "Posits and Reality"

Having noted that man has no evidence for the existence of bodies beyond the fact that their assumption helps him organize experience, we should have done well, instead of disclaiming evidence for the existence of bodies, to conclude: such then, at bottom, is what evidence is, both for ordinary bodies and for molecules. . . .

So for Quine, our belief in the best scientific theories is continuous with our own sensory data, in the manner of his scientific holism. We have no reason to pragmatically believe in molecules or abstract objects any more than we should only pragmatically believe in the chair we're sitting in.

Interesting, thanks I never read this piece. Well, his commitment to both pragmatism and Skinner's behaviorism is evident here.

Quine definitely hushed down his commitment to old-school American Pragmatism, or maybe it's the scholarship that portrays him as a direct continuation of Logical Positivism to create some linear narrative to the history of Analytic Philosophy.

[palandio]

I would say yes.

If nature exists outside the human mind, then there is that kind of structure in it that is as mathematical as life is biological and matter and force are physical. Now there may be different mathematical formalizations that exist only in our minds, but there is an underlying reality. And going even further I don't see why there would be reality behind 2- and 3-dimensional geometry, but not behind higher-dimensional geometry. I mean 6-dimensional geometry is used to describe the position of physical objects like a tool (5-dimensional if the tool is rotating), if we want to describe the phase space we're already at 12 dimensions, 11 if energy is conserved. And then there probably is a reality/truth behind n-dimensional geometry in general. And behind topology, and so on.

But there are reasons why I'm not a philosopher. I'm wilfully ignorant when it comes to philosophy of Mathematics, it makes my head ache. I'm just a simple man trained in a mathematical discipline that is usually considered Applied Math although it has increasingly moved into the abstract direction during the last decades. If someone put a gun to my head and asked if Mathematics were a Naturwissenschaft or a Geisteswissenschaft, I would say Naturwissenschaft, unlike many pure Mathematicians among whom it has become fashionable to consider Mathematics a Geisteswissenschaft, but they don't know what a Geisteswissenschaft really is, I think.

(If this sounded too sarcastic for you, I actually enjoy discussing with philosophers a lot, but I also enjoy just doing and applying math without constantly reflecting about the philosophical essence of what I'm doing.)

That's an interesting position and it sounds similar to Paul Benacerraf's structuralism in mathematics and perhaps entity realism in sciences.

It's that platonic theories like numbers exist as some abstract ideas hovering above the ultimate reality that usually gives ire to anti-realists like me, not the softer realism.

I find it odd when someone says the number 2 as an object must exist in order for us to use it in math. It seems unnecessary for me as there is nothing fundamental about numbers. There is a sense in which the number two appears in:
there is an (x), there is a (y) such that: P applies to x & P applies to y & x=/=y
without being an object

Thank you for the suggestions. I will try to read more about Benaceraff, entity realism, structural realism, etc.

Platonic realism and generally the more extreme forms of realism seem a bit naive to me. But still I would find it discomforting to consider mathematics to exist just in our minds, just because we need to use abstraction. If you think about it, 'life', 'cell', 'organism', 'animal', etc. are abstractions (or universals?) as much as '2' or 'space' are. And scientific realism in the natural sciences outside mathematics is still dominant, although maybe a higher dose of healthy skepticism would be adequate.

The irony is that in the field in which I have the most experience, i.e. probability, I'm agnostic about the reality of random itself. Not only can probability theory be put on a purely axiomatic foundation, but it can be applied in any context where you quantify lack of information (the deFinettian approach). At the same time I like to think that there is e.g. a real structure outside the human mind to which the Central Limit Theorem (and the various versions of it) gives us an insight.

One last thing about objects. Years ago a senior student asked one of my geometry professors: "Excuse me, what is geometry according to you? It's all about objects, isn't it?" And the professor said: "No, no, no. It's about spaces with structures on them and their relations." And this in my opinion applies to most of mathematics. Mathematical objects alone are almost meaningless constructions of our mind. It is their interactions that hopefully can give us a glimpse on interactions that are going on outside our minds.

Do you as a mathematician feel intuitionist logic and category theory to be "realer" and better foundation to mathematics than standard logic and set theory? Category theory at least on paper tries to ground relation in spatial terms

I think that convenience plays a role in the common preference for standard logic and set theory. In my opinion you can interpret the following famous quote from Hilbert in that way: "From the paradise, that Cantor created for us, no-one shall be able to expel us." Most of mathematics can be "re-created" using intuitionist logic, although it often becomes a bit more tiresome. And the statements that really rely on the law of excluded middle or, God beware, the axiom of choice, honestly I personally don't put too much faith into them. In practice it is still often convenient to use e.g. the law of excluded middle and in most situations it doesn't feel wrong. I'm sympathetic to category theory because it formalizes a lot of what I think is at the core of mathematics. Sadly probability is often ignored as an application field by category theorists that come from algebra, topology or logic, whereas probabilists use categories and category theory but never say the word 'category theory'.

[muon2]

As a practicing scientist with a degree in mathematics I can give you my view of the reality of mathematical expressions independent of specific philosophical interpretations.

In science math is our primary tool kit to understand relationships between observable physical phenomena. Let me use the analogy of the scientist measuring the force of friction between two sufaces and its impact on motion as a carpenter building a bird house. Like the scientist the carpenter has both raw materials and a tool box to fashion new relationships from those raw materials. And both can use simple tools or more sophisticated ones (consider a hammer and saw versus a computer controlled milling and joining machine as simple arithmetic versus vector calculus). In one case there is a relationship describing frictional force and in the other a relationship between pieces of wood we callit a bird house. If the tools used are simple the relationship may appear simple as well and perhaps less pleasing due to a lack of precision, but the relationship is made nonetheless.

So do I think the carpenters tools are real? Yes, but I wouldn't confuse them with a product of the carpenter's craft. Do I think that the mathematics used in science is real? Yes, but I wouldn't confuse it with the physical relations deduced by their use. I am happy to employ greater levels of abstract mathematics as need be to construct the relationships I seek based on my need for precision, that doesn't make those more abstract levels less real than the simple ones, nor does it make even those simple levels of mathematics into physical entities.

Make of that what you will.

[Torie]

As a practicing scientist with a degree in mathematics I can give you my view of the reality of mathematical expressions independent of specific philosophical interpretations.

In science math is our primary tool kit to understand relationships between observable physical phenomena. Let me use the analogy of the scientist measuring the force of friction between two sufaces and its impact on motion as a carpenter building a bird house. Like the scientist the carpenter has both raw materials and a tool box to fashion new relationships from those raw materials. And both can use simple tools or more sophisticated ones (consider a hammer and saw versus a computer controlled milling and joining machine as simple arithmetic versus vector calculus). In one case there is a relationship describing frictional force and in the other a relationship between pieces of wood we callit a bird house. If the tools used are simple the relationship may appear simple as well and perhaps less pleasing due to a lack of precision, but the relationship is made nonetheless.

So do I think the carpenters tools are real? Yes, but I wouldn't confuse them with a product of the carpenter's craft. Do I think that the mathematics used in science is real? Yes, but I wouldn't confuse it with the physical relations deduced by their use. I am happy to employ greater levels of abstract mathematics as need be to construct the relationships I seek based on my need for precision, that doesn't make those more abstract levels less real than the simple ones, nor does it make even those simple levels of mathematics into physical entities.

Make of that what you will.

How much time do you spend in this place, and how "real" is it?

https://en.wikipedia.org/wiki/Four-dimensional_space

The only higher mathematics I ever found practical in my bottom dweller existence was simultaneous algebraic equations, when using NOI's of real estate to calculate value, where the higher the value the higher the property tax, so you have two variables influencing each other. Now with my brain fading, I just use a spreadsheet until the two values merge. Sad!

[muon2]

As a practicing scientist with a degree in mathematics I can give you my view of the reality of mathematical expressions independent of specific philosophical interpretations.

In science math is our primary tool kit to understand relationships between observable physical phenomena. Let me use the analogy of the scientist measuring the force of friction between two sufaces and its impact on motion as a carpenter building a bird house. Like the scientist the carpenter has both raw materials and a tool box to fashion new relationships from those raw materials. And both can use simple tools or more sophisticated ones (consider a hammer and saw versus a computer controlled milling and joining machine as simple arithmetic versus vector calculus). In one case there is a relationship describing frictional force and in the other a relationship between pieces of wood we callit a bird house. If the tools used are simple the relationship may appear simple as well and perhaps less pleasing due to a lack of precision, but the relationship is made nonetheless.

So do I think the carpenters tools are real? Yes, but I wouldn't confuse them with a product of the carpenter's craft. Do I think that the mathematics used in science is real? Yes, but I wouldn't confuse it with the physical relations deduced by their use. I am happy to employ greater levels of abstract mathematics as need be to construct the relationships I seek based on my need for precision, that doesn't make those more abstract levels less real than the simple ones, nor does it make even those simple levels of mathematics into physical entities.

Make of that what you will.

How much time do you spend in this place, and how "real" is it?

https://en.wikipedia.org/wiki/Four-dimensional_space

The only higher mathematics I ever found practical in my bottom dweller existence was simultaneous algebraic equations, when using NOI's of real estate to calculate value, where the higher the value the higher the property tax, so you have two variables influencing each other. Now with my brain fading, I just use a spreadsheet until the two values merge. Sad!

I spent some time teaching the 4 dimensional physics of relativity to young minds last semester. I noted that without those needed corrections many of our precision technologies, such as GPS, wouldn't be of any use.

[vitoNova]

All mathematics is derived from the five senses of an unremarkable African ape species in an otherwise unremarkable planet in an otherwise unremarkable star in an unremarkable arm of an unremarkable galaxy. 

Yes.  I've followed the footsteps of Heisenberg in the North Sea.  I've had my fair share of drunken debates in Club 1900 and Cave 54 amongst Heidelberg Univerität physics and geography students.

But yeah, y'all wrong bruh.   The maths is human construction. 

[Hnv1]

All mathematics is derived from the five senses of an unremarkable African ape species in an otherwise unremarkable planet in an otherwise unremarkable star in an unremarkable arm of an unremarkable galaxy.  

Yes.  I've followed the footsteps of Heisenberg in the North Sea.  I've had my fair share of drunken debates in Club 1900 and Cave 54 amongst Heidelberg Univerität physics and geography students.

But yeah, y'all wrong bruh.   The maths is human construction.  

That the signs of math are constructions by us is obvious. There is nothing in "1"/ "one" / "I" or any representation device to intrinsically make it real. Nor is our methods of numbering (1-10) necessary. Even where our signs act like Peirce's icons.  

The question is whether my assertion that "I have two socks on right now" taps into something that is invariant between its representations. i.e., that everything that represents, will tap into some equivalent description. Would our alien overlords from the remarkable intergalactic empire represent nature in some similar mathematical way, or at least commensurable, or at least syntactically commensurable, way?

I don't think hand-waiving reductionism of our formalism to phenomenology does the heavy lifting you think it does here.

[Sestak]

I expect my thoughts on this will come as a surprise to no one; mathematics is a set of universal truths/tautologies based on axiom sets; they are not real in the sense that if nothing was real they would still hold true. Even in a universe which operated under a completely different mathematical system, our mathematics would still hold entirely true - it would just also be rather useless because the axioms couldn't be applied to anything in the real world. But the statements of "If [x axioms] then [y result] would still be true.

Which areas of mathematics we consider useful and decide to devote time to study is absolutely a subjective decision on our own part, though a subjective decision that is (largely) determined by the operation of the real world.

[palandio]

All mathematics is derived from the five senses of an unremarkable African ape species in an otherwise unremarkable planet in an otherwise unremarkable star in an unremarkable arm of an unremarkable galaxy. 

Yes.  I've followed the footsteps of Heisenberg in the North Sea.  I've had my fair share of drunken debates in Club 1900 and Cave 54 amongst Heidelberg Univerität physics and geography students.

But yeah, y'all wrong bruh.   The maths is human construction. 

Isn't that true of other sciences like physics and biology, too? Don't we use the term "physics" to describe both the science and the reality (whatever that is) it tries to make sense of? Or does our need to perceive reality through our five(?) senses cut us off from reality completely? Then even the subjects of science would only exist in our minds with no meaningful connection to reality (if there is anything like reality at all).

[vitoNova]

All mathematics is derived from the five senses of an unremarkable African ape species in an otherwise unremarkable planet in an otherwise unremarkable star in an unremarkable arm of an unremarkable galaxy. 

Yes.  I've followed the footsteps of Heisenberg in the North Sea.  I've had my fair share of drunken debates in Club 1900 and Cave 54 amongst Heidelberg Univerität physics and geography students.

But yeah, y'all wrong bruh.   The maths is human construction. 

Isn't that true of other sciences like physics and biology, too? Don't we use the term "physics" to describe both the science and the reality (whatever that is) it tries to make sense of? Or does our need to perceive reality through our five(?) senses cut us off from reality completely? Then even the subjects of science would only exist in our minds with no meaningful connection to reality (if there is anything like reality at all).

Keep in mind that I believe mathematics is quite possibly the most vitally important of all "human constructs" and that yes indeed, it taps very, very close into the peripheries of objective reality.  Even if it is perpetually doomed to fail in understanding what reality truly is.

Also in case you haven't noticed, I'm a YUUGE fan of simulation theory.  And therefore any hypothetical advanced extraterrestrial species that is also well-versed in "1 apple and another apple makes 2 apples" will still be forever constrained by its biological brain.

Also as I'm sure you've noticed, I don't have one iota of a STEM background and my BA was in a quite decidedly non-STEM field.   I still found good-natured debates with German physics and philosophy students over several pints of Hefeweißen quite enlightening.   You simply don't get that here in the states.

[muon2]

All mathematics is derived from the five senses of an unremarkable African ape species in an otherwise unremarkable planet in an otherwise unremarkable star in an unremarkable arm of an unremarkable galaxy. 

Yes.  I've followed the footsteps of Heisenberg in the North Sea.  I've had my fair share of drunken debates in Club 1900 and Cave 54 amongst Heidelberg Univerität physics and geography students.

But yeah, y'all wrong bruh.   The maths is human construction. 

Isn't that true of other sciences like physics and biology, too? Don't we use the term "physics" to describe both the science and the reality (whatever that is) it tries to make sense of? Or does our need to perceive reality through our five(?) senses cut us off from reality completely? Then even the subjects of science would only exist in our minds with no meaningful connection to reality (if there is anything like reality at all).

Keep in mind that I believe mathematics is quite possibly the most vitally important of all "human constructs" and that yes indeed, it taps very, very close into the peripheries of objective reality.  Even if it is perpetually doomed to fail in understanding what reality truly is.

Also in case you haven't noticed, I'm a YUUGE fan of simulation theory.  And therefore any hypothetical advanced extraterrestrial species that is also well-versed in "1 apple and another apple makes 2 apples" will still be forever constrained by its biological brain.

Also as I'm sure you've noticed, I don't have one iota of a STEM background and my BA was in a quite decidedly non-STEM field.   I still found good-natured debates with German physics and philosophy students over several pints of Hefeweißen quite enlightening.   You simply don't get that here in the states.

I disagree. I still see that type of interaction at small liberal arts colleges, usually without the Hefeweißen since many of the students are under 21.