Mathematical Procedures and Incommensurability
I. Procedure and Representation
We can use numbers to perform calculations without having to stipulate that each number refers to anything outside of our mathematical operations. Number systems are tools for counting and performing other arithmetical functions. We can define arithmetic procedurally, and avoid wondering what sort of existence numbers might have on their own, perhaps in some Platonic realm. Numbers are symbols used to represent mathematical procedures.
I used to think that the existence of irrational numbers posed a problem for this view. To define rational numbers, we say they can be represented as a fraction between m and n (where m and n are not both divisible by two). Irrational numbers are defined as numbers which cannot be represented as fractions in this way. They seem to point to something beyond comprehension, beyond the possibility of finite containment.
Indeed, the fact is, we have symbols for irrational numbers; the numbers themselves are the referent. So in what sense can we say that irrational numbers are representations of procedures, if the numbers in question cannot be fully represented in any finite space?
This is a fundamental logical problem, or so it has been claimed.
II. Infinity and Impossible Objects
A long time ago I thought it was a good idea to define "infinity" as "the relationship between a discrete point and a continuous line." I don't know why that definition occurred to me, or why I liked it. Probably because of something to do with Zeno's paradoxes.
The sense is thus: A discrete point has no extension. A line has extension. Therefore, no matter how many discrete points you attempt to connect, you will never get a line. And no matter how many discrete points you attempt to extract from a line, you will never decrease its length. So, there is an infinite number of discrete points on any line, and a line is infinitely longer than any series of discrete points.
As a definition, this may not be very useful. But as a way of thinking about infinity, I find it very interesting. And implicit in this approach is an incommensurability; namely, that between extended and unextended objects.
Consider pi, which defines the relationship between a circle's radius and its circumference, as well as between its radius and its area. Pi is an irrational number. Why?
Well, one way to look at it is to consider how we define a circle as such. We define a circle as the set of all points on a linear plane equidistant to a single point. Yet, in this case, a circle is defined in terms of discrete points, which have no extension. A circle is a continuous line. So we have an incommensurability, and this indicates a geometrical impossibility. Pi is irrational because it is impossible to produce a perfect circle.
This is not a human limitation. It is a geometrical fact.
Yet, in some sense, mathematicians say that pi exists. The word "pi" represents something real, even if it cannot be calculated completely. We can calculate it to any arbitrary degree of accuracy. The question is, what does "accuracy" mean here? What do our calculations signify?
From the procedural point of view, the calculations signify steps in our attempt to generate a circle.
Consider how we might go about constructing a circle in the real world. We might try any number of ways. The irrationality of pi indicates that, no matter what method we use, there will always be a better way. It's not that we are getting closer and closer to the true value of pi. Rather, it is that we are getting closer and closer to a perfect circle, even though such an object cannot exist. We can approach impossible objects with infinite precision; we just cannot make them. So, pi is irrational. This is a geometrical fact about circles, and not about whatever might be trying to generate them.
III. Geometry and Arithmetic
Irrational numbers are found when we attempt to produce arithmetical models of impossible geometrical operations. Irrational numbers indicate a tension between geometry and arithmetic.
I recently explored this idea by trying to find a geometrical procedure for creating a single perfect cube out of the parts of three identical cubes. If it is possible to create such a cube, then the cube root of 3 must be calculatable. Since the cube root of 3 is irrational, I suppose that one cannot, even in theory, combine three cubes into a single one.
Today I thought of another example: the square root of five. Consider a square with sides length 2 meters. We would thus say the area of this square is 4 square meters. I suggest that it is theoretically impossible, by any means imaginable, to increase the size of such a cube by exactly twenty-five percent, arriving at a square with an area of 5 square meters.
Then I did a couple Web searches on irrational numbers, and I was happy to find this page, which supports my understanding. It says irrational numbers surface at the intersection of arithmetic and geometry, and that they indicate incommensurability. The author of that site (Laurence Spector, a math instructor at a community college in New York) claims here that there is a "fundamental logical problem" concerning the existence of irrational numbers. He suggests that the tension between arithmetic and geometry exists because it is impossible to name or measure every length.
I think there is something wrong with that explanation. Are we to take it that irrational numbers represent unmeasurable lengths?
Now, Spector notes that in order for irrational numbers to be considered numbers at all, we must have a procedure to name, or measure, them to any arbitrary degree. That is, to say that pi is a number, we must have some procedure which allows us to place it on the continuum of Real numbers. This doesn't mean we designate a specific spot for it; rather, it means that we can place it between two rational numbers--two numbers which we know how to measure.
But why does Spector conclude that the existence of irrational numbers means that not every length is measurable?
His assumption, apparently, is that irrational numbers represent lengths. This is the problem.
Numbers do not represent lengths. People may represent lengths by using numbers. (Similarly, words do not refer to things. People refer to things using words.)
Am I referring to a specific length when I say "the square root of two meters?" No, I don't think so. Not unless somebody told me that some particular length was the square root of two meters. But calling any length "the square root of two meters" can only be arbitrarily justified, and not according to any definitive standard of measurement. There is no way to determine that any length is equal to the square root of two meters, but we can always say it's close enough. This doesn't mean that any particular length is unmeasurable. It means "the square root of two meters" doesn't pick out a particular length at all.
When we produce a longer calculation of pi, we are not getting a closer approximation to the true relationship between a circle's radius and its circumference. Rather, we getting a more detailed standard of measurement.
More detailed. Not more precise or more accurate.
Since there is no end to the possible digits you can place to the right of the decimal point, there is no sense in claiming that you're ever getting closer to the end. The longer the calculation, the more information we have; but this doesn't make the information better in any purely mathematical sense. It is only better if you have some purpose, some use, for all of those numbers.
IV. Conclusion
I see no sense in talking about unknowable, unmeasurable, or infinitely improbable lengths. Lengths are defined in terms of operations, procedures. We get numbers like "the square root of two" because we are devising procedures which have no geometrical correlates. There is no procedure for producing a perfect circle in geometry. There is no procedure for increasing a square by twenty-five percent, or of combining three identical cubes into one. These are theoretical limitations, not practical obstacles.
When we say irrational numbers are real, we mean that our procedures for calculating them are valid. I have no issue with that point. The question is not whether or not the calculations are valid; the question is what they mean.
I don't know if Spector's interpretation is common among philosophers or mathematicians. However, it seems to me that the supposed "logical problem" is only a problem of interpretation, of thinking that numbers indicate anything other than the procedures we have for calculating them.
We can use numbers to perform calculations without having to stipulate that each number refers to anything outside of our mathematical operations. Number systems are tools for counting and performing other arithmetical functions. We can define arithmetic procedurally, and avoid wondering what sort of existence numbers might have on their own, perhaps in some Platonic realm. Numbers are symbols used to represent mathematical procedures.
I used to think that the existence of irrational numbers posed a problem for this view. To define rational numbers, we say they can be represented as a fraction between m and n (where m and n are not both divisible by two). Irrational numbers are defined as numbers which cannot be represented as fractions in this way. They seem to point to something beyond comprehension, beyond the possibility of finite containment.
Indeed, the fact is, we have symbols for irrational numbers; the numbers themselves are the referent. So in what sense can we say that irrational numbers are representations of procedures, if the numbers in question cannot be fully represented in any finite space?
This is a fundamental logical problem, or so it has been claimed.
II. Infinity and Impossible Objects
A long time ago I thought it was a good idea to define "infinity" as "the relationship between a discrete point and a continuous line." I don't know why that definition occurred to me, or why I liked it. Probably because of something to do with Zeno's paradoxes.
The sense is thus: A discrete point has no extension. A line has extension. Therefore, no matter how many discrete points you attempt to connect, you will never get a line. And no matter how many discrete points you attempt to extract from a line, you will never decrease its length. So, there is an infinite number of discrete points on any line, and a line is infinitely longer than any series of discrete points.
As a definition, this may not be very useful. But as a way of thinking about infinity, I find it very interesting. And implicit in this approach is an incommensurability; namely, that between extended and unextended objects.
Consider pi, which defines the relationship between a circle's radius and its circumference, as well as between its radius and its area. Pi is an irrational number. Why?
Well, one way to look at it is to consider how we define a circle as such. We define a circle as the set of all points on a linear plane equidistant to a single point. Yet, in this case, a circle is defined in terms of discrete points, which have no extension. A circle is a continuous line. So we have an incommensurability, and this indicates a geometrical impossibility. Pi is irrational because it is impossible to produce a perfect circle.
This is not a human limitation. It is a geometrical fact.
Yet, in some sense, mathematicians say that pi exists. The word "pi" represents something real, even if it cannot be calculated completely. We can calculate it to any arbitrary degree of accuracy. The question is, what does "accuracy" mean here? What do our calculations signify?
From the procedural point of view, the calculations signify steps in our attempt to generate a circle.
Consider how we might go about constructing a circle in the real world. We might try any number of ways. The irrationality of pi indicates that, no matter what method we use, there will always be a better way. It's not that we are getting closer and closer to the true value of pi. Rather, it is that we are getting closer and closer to a perfect circle, even though such an object cannot exist. We can approach impossible objects with infinite precision; we just cannot make them. So, pi is irrational. This is a geometrical fact about circles, and not about whatever might be trying to generate them.
III. Geometry and Arithmetic
Irrational numbers are found when we attempt to produce arithmetical models of impossible geometrical operations. Irrational numbers indicate a tension between geometry and arithmetic.
I recently explored this idea by trying to find a geometrical procedure for creating a single perfect cube out of the parts of three identical cubes. If it is possible to create such a cube, then the cube root of 3 must be calculatable. Since the cube root of 3 is irrational, I suppose that one cannot, even in theory, combine three cubes into a single one.
Today I thought of another example: the square root of five. Consider a square with sides length 2 meters. We would thus say the area of this square is 4 square meters. I suggest that it is theoretically impossible, by any means imaginable, to increase the size of such a cube by exactly twenty-five percent, arriving at a square with an area of 5 square meters.
Then I did a couple Web searches on irrational numbers, and I was happy to find this page, which supports my understanding. It says irrational numbers surface at the intersection of arithmetic and geometry, and that they indicate incommensurability. The author of that site (Laurence Spector, a math instructor at a community college in New York) claims here that there is a "fundamental logical problem" concerning the existence of irrational numbers. He suggests that the tension between arithmetic and geometry exists because it is impossible to name or measure every length.
I think there is something wrong with that explanation. Are we to take it that irrational numbers represent unmeasurable lengths?
Now, Spector notes that in order for irrational numbers to be considered numbers at all, we must have a procedure to name, or measure, them to any arbitrary degree. That is, to say that pi is a number, we must have some procedure which allows us to place it on the continuum of Real numbers. This doesn't mean we designate a specific spot for it; rather, it means that we can place it between two rational numbers--two numbers which we know how to measure.
But why does Spector conclude that the existence of irrational numbers means that not every length is measurable?
His assumption, apparently, is that irrational numbers represent lengths. This is the problem.
Numbers do not represent lengths. People may represent lengths by using numbers. (Similarly, words do not refer to things. People refer to things using words.)
Am I referring to a specific length when I say "the square root of two meters?" No, I don't think so. Not unless somebody told me that some particular length was the square root of two meters. But calling any length "the square root of two meters" can only be arbitrarily justified, and not according to any definitive standard of measurement. There is no way to determine that any length is equal to the square root of two meters, but we can always say it's close enough. This doesn't mean that any particular length is unmeasurable. It means "the square root of two meters" doesn't pick out a particular length at all.
When we produce a longer calculation of pi, we are not getting a closer approximation to the true relationship between a circle's radius and its circumference. Rather, we getting a more detailed standard of measurement.
More detailed. Not more precise or more accurate.
Since there is no end to the possible digits you can place to the right of the decimal point, there is no sense in claiming that you're ever getting closer to the end. The longer the calculation, the more information we have; but this doesn't make the information better in any purely mathematical sense. It is only better if you have some purpose, some use, for all of those numbers.
IV. Conclusion
I see no sense in talking about unknowable, unmeasurable, or infinitely improbable lengths. Lengths are defined in terms of operations, procedures. We get numbers like "the square root of two" because we are devising procedures which have no geometrical correlates. There is no procedure for producing a perfect circle in geometry. There is no procedure for increasing a square by twenty-five percent, or of combining three identical cubes into one. These are theoretical limitations, not practical obstacles.
When we say irrational numbers are real, we mean that our procedures for calculating them are valid. I have no issue with that point. The question is not whether or not the calculations are valid; the question is what they mean.
I don't know if Spector's interpretation is common among philosophers or mathematicians. However, it seems to me that the supposed "logical problem" is only a problem of interpretation, of thinking that numbers indicate anything other than the procedures we have for calculating them.
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