Alternative math

by Neil Rickert

Jerry Coyne posted this video at his blog:

Presumably Coyne was making a point about alternative truths.  The video clip is quite exaggerated.  I don’t expect anything like that to actually happen.   But, of course, exaggeration is often a good way of making a point.

However, I see the video as also making a very different point.  And that’s what this post is about.

Mathematics as culture

To be a mathematician is to be a member of a culture.  To do mathematics, is to follow the conventions of that mathematics culture.

That 2+2 = 4 is a convention within the mathematics culture.  It has been argued that mathematics is entirely a matter of convention (or cultural practices).  Quine argues against that in his paper “Truth by Convention”, which I suppose I should discuss some day.

So yes, you could have an alternative-math culture where the convention is that 2+2 = 22.  You would not be able to use alternative-math in the same way that you use standard math.  In alternative-math, addition would not be connected with counting unless you also came up with an alternative form of counting.

We follow the conventions of standard mathematics, because they have proved very useful.  The conventions themselves were developed for their usefulness.  So the conventions of mathematics are not random.  They are chosen for their usefulness.  I very much doubt that alternative-math could be developed into as useful a system of conventions.  But, strictly speaking, I am only guessing when I say that.

Conventions are important

The take away from this should be to recognize the importance of social conventions.  Our traffic laws are conventions.  The rules of baseball are conventions.  Our banking systems depend on conventions.  Our markets (stock markets, food markets, etc) all follow social conventions that have been honed over many years.

The importance of social conventions has, I think, been under appreciated.

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11 Comments to “Alternative math”

  1. Do you beleive that our universe has objective mathematical laws “built”-into it?

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  2. Meant “believe”. Sorry I cannot spell…..
    Also, maybe better to ask if you think there are ANY objective mathematical laws?

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    • The difficulty here is that people disagree over the meaning of “objective”.

      From one point of view, mathematics is purely subjective, because it takes place in the mind. From a different point of view, mathematics is objective because there is complete agreement between people — and that makes mathematics the most objective thing that we have.

      I’m inclined to go with that second viewpoint.

      Most mathematicians would agree that the complex numbers are an invented number system. But, once they were invented, Euler’s formula e^{i\pi} + 1 = 0 is a remarkable discovery.

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  3. “The importance of social conventions has, I think, been under appreciated.”

    I took you to be saying that (many) social conventions (like maths) are as they are because they are useful – and they are useful because they are not arbitrary. They mesh with a reality more encompassing than social reality.

    But then in a comment you said: “From a different point of view, mathematics is objective because there is complete agreement between people — and that makes mathematics the most objective thing that we have.”

    I would call this kind of agreement intersubjectivity. And I think I would want to claim more for basic science and maths. I.e. that it reflects or meshes with the physical world, not just the social/psychological world.

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    • I would call this kind of agreement intersubjectivity.

      Fair enough.

      As I see it, intersubjectivity is the only kind of objectivity that we can ever have.

      And I think I would want to claim more for basic science and maths. I.e. that it reflects or meshes with the physical world, not just the social/psychological world.

      The point with mathematics, is that it does not mesh with the physical world. It is very much an idealization.

      Counting, in the physical world, looks like the “hanging chads” problem we saw in Florida, with the 2000 presidential election. Arithmetic completely idealizes that. If we were concerned with actual counting, then we would never get as far as the prime number theorem.

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  4. Well as I see it the world/ the cosmos is a certain way and some but not all mathematical idealizations are useful to describe it.

    But I was really talking about science more than maths. That’s where I really want some kind of objectivity. I am open to the idea that maths is always an idealization and only describes reality imperfectly. (You could imagine an advanced science without maths as we know it.)

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    • Well as I see it the world/ the cosmos is a certain way …

      I have two or three posts where I have argued against the idea that the world is a certain way.

      Yes, it is trivially obvious that the world is a certain way. And for thousands of years, it was trivially obvious that the sun took a daily trip around the earth. But that no longer seems so certain.

      The trouble is that all of the “certain ways” that we can think of, are human inventions. So “there’s a certain way …” seems meaningless.

      It looks to me as if “there’s a certain way …” is used to insist on a theistic conception of truth but without committing oneself to theism.

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      • “Yes, it is trivially obvious that the world is a certain way. And for thousands of years, it was trivially obvious that the sun took a daily trip around the earth. But that no longer seems so certain.”

        I was thinking more in terms of the conviction that drives scientists to propose hypotheses so that they may be *tested* by observation and experiment. Of course when you have movement it can be seen from various points of view.

        “The trouble is that all of the “certain ways” that we can think of, are human inventions. So “there’s a certain way …” seems meaningless.”

        A well-established theory is indeed a human invention but it is one that is consistent with multiple observations and perhaps the basis for successful technologies. It therefore tells us something about how the world is/works.

        “It looks to me as if “there’s a certain way …” is used to insist on a theistic conception of truth but without committing oneself to theism.”

        Well if you are not insisting on theism it’s not really theistic, I would have thought. But I see your point and it is one that is often made. But what does it amount to? That there are certain similarities? You could claim perhaps that the view I am putting derives historically from a theistic view (but even if so, so what?). Or that it is similar in some respects to a theistic view. But, again, so what? The question is: is it a plausible view (if you assume no God)?

        A moral realist might need to revise his metaethics if he stops believing in God, but a scientific realist, no. I don’t see the problem. (The so-called “laws” of nature are not like moral imperatives.)

        I have read quite a lot on this and would be happy to read (or reread) what you have written on it, but nothing I have read attacking scientific realism has seemed compelling to me.

        I might add that I think that this is a very important issue and anything I can do to clarify my own or others’ thinking on it I consider very worthwhile.

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        • A well-established theory is indeed a human invention but it is one that is consistent with multiple observations and perhaps the basis for successful technologies. It therefore tells us something about how the world is/works.

          I agree with that.

          Well if you are not insisting on theism it’s not really theistic, I would have thought. But I see your point and it is one that is often made. But what does it amount to?

          By “a theistic conception of truth”, I mean a view that insists that truth is completely external to humans, similar to what you would expect with a central truth giver (such as a monotheistic deity). Many people seem to have that sort of view of truth. By contrast, I see truth as a human linguistic construct.

          I have read quite a lot on this and would be happy to read (or reread) what you have written on it, but nothing I have read attacking scientific realism has seemed compelling to me.

          I think of myself as a scientific realist. But some philosophers apparently think I’m an anti-realist.

          My own view is that there is a human independent reality that scientists study and attempt to describe. And that’s about what I take “scientific realism” to imply. But the philosophers who criticize me for this want to also assert something about the truth of those scientific descriptions. I see scientists as pragmatists, concerned mainly with the usefulness of their descriptions, rather than with whether those descriptions are true.

          I can illustrate that with an example. People who take what I am calling a theistic conception of truth will most likely say that heliocentrism is true. They will perhaps admit that they could be mistaken. But they insist that there is a single unitary answer as to whether heliocentrism is true, even if their current belief might be mistaken.

          By contrast, I take heliocentrism to be a pragmatic social convention (mostly a scientific convention). As best I can tell there are no standards by which we could judge whether heliocentrism is true, other than the standard set by heliocentrism itself as a social convention.

          I’m really not trying to be argumentative about truth just for the sake of arguing. Rather, I’m studying human cognition. And I am trying to point out the importance of developing pragmatic social conventions which can then be used as standards. The view “that there is a certain way …” misses a great deal of what a cognitive system has to do, and thus makes human cognition look more magical and mystical than it actually is.

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          • “I think of myself as a scientific realist…”

            Well we are probably not too far apart.

            “My … view is that there is a human independent reality that scientists study and attempt to describe. And that’s about what I take “scientific realism” to imply.”

            Okay.

            “But the philosophers who criticize me for this want to also assert something about the truth of those scientific descriptions. I see scientists as pragmatists, concerned mainly with the usefulness of their descriptions, rather than with whether those descriptions are true.”

            “True” is a word which (as far as I am concerned) is quite innocent and unobjectionable if it is used in an ordinary (non philosophical) way. I tend to take a deflationary approach. To say that an assertion is true is doing nothing more than making the assertion in question.

            “By “a theistic conception of truth”, I mean a view that insists that truth is completely external to humans, similar to what you would expect with a central truth giver (such as a monotheistic deity). Many people seem to have that sort of view of truth. By contrast, I see truth as a human linguistic construct.”

            You are not using the word “truth” in the ordinary way here. (“[T]ruth” [?] being “external”; “central truth giver”, etc..) You are using it in a kind of philosophical way – and so (as I see it) are creating an artificial problem.

            “People who take what I am calling a theistic conception of truth will most likely say that heliocentrism is true. They will perhaps admit that they could be mistaken. But they insist that there is a single unitary answer as to whether heliocentrism is true, even if their current belief might be mistaken.”

            I don’t think this is a good or typical example because the movement of the objects in the solar system can be conceptualized in different ways. The standard heliocentric picture seems the simplist and most economical or easy to describe.

            “… I’m studying human cognition. And I am trying to point out the importance of developing pragmatic social conventions which can then be used as standards.”

            As standards or to develop standards? You say “pragmatic social conventions”. Why not just social conventions? Or practices?

            “The view “that there is a certain way …” misses a great deal of what a cognitive system has to do, and thus makes human cognition look more magical and mystical than it actually is.”

            You are picking me up on that phrase. But when I said that the world/cosmos is a certain way I did not mean anything more than that (as you yourself put it) there is a human-independent reality which scientists seek to study and describe.

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          • You say “pragmatic social conventions”. Why not just social conventions? Or practices?

            Well, yes, “pragmatic” is mostly redundant. Conventions are typically pragmatic in some way. But people often hear the expression “arbitrary conventions” and they mistakenly take “arbitrary” to imply “random”. So I have found that it helps to avoid such confusion if I explicitly mention “pragmatic.”

            You are picking me up on that phrase. But when I said that the world/cosmos is a certain way I did not mean anything more than that (as you yourself put it) there is a human-independent reality which scientists seek to study and describe.

            Fair enough. Then I think we mostly agree.

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