Indeed, even though when you use a Lossy-Correspondence/Compression Theory of Truth, abstract objects become perfectly sensible as descriptions of regularities in concrete objects.
Not really, because most maths is unphysical, ie physics is picking out the physically applicable parts of maths, ie the rest has nothing to correspond to.
If I remember my Lakoff & Núñez correctly, they were arguing that even the most abstract and un-physical-seeming of maths is constructed on foundations that derive from the way we perceive the physical world.
Let me pick up the book again… ah, right. They define two kinds of conceptual metaphor:
Grounding metaphors yield basic, directly grounded ideas. Examples: addition as adding objects to a collection, subtraction as taking objects away from a collection, sets as containers, members of a set as objects in a container. These usually require little instruction.
Linking metaphors yield sophisticated ideas, sometimes called abstract ideas. Examples: numbers as points on a line, geometrical figures as algebraic equations, operations on classes as algebraic operations. These require a significant amount of explicit instruction.
Their argument is that for any kind of abstract mathematics, if you trace back its origin for long enough, you finally end up at some grounding and linking metaphors that have originally been derived from our understanding of physical reality.
As an example of the technique, they discuss the laws of arithmetic as having been derived from four grounding metaphors: Object Collection (if you put one and one physical objects together, you have a collection of two objects), Object Construction (physical objects are made up of smaller physical objects; used for understanding expressions like “five is made up of two plus three” or “you can factor 28 into 7 times 4″), Measuring Stick (physical distances correspond to numbers; gave birth to irrational numbers, when the Pythagorean theorem was used to prove their existence by assuming that there’s a number that corresponds to the length of the hypotenuse), and Motion Along A Path (used in the sixteenth century to invent the concept of the number line, and the notion of a number as lying between two other numbers).
Now, they argue that these grounding metaphors, each by themselves, are not sufficient to define the laws of arithmetic for negative numbers. Rather you need to combine them into a new metaphor that uses parts of each, and then define your new laws in terms of that newly-constructed metaphor.
Defining negative numbers is straightforward using these metaphors: if you have the concept of a number line, you can define negative numbers as “point-locations on the path on the side opposite the origin from positive numbers”, so e.g. −5 is the point five steps to the left of the origin point, symmetrical to +5 which is five steps to right of the origin point.
Next we can use Motion Along A Path to define addition and subtraction: adding positive numbers is moving towards the right, addition of negative numbers is moving towards the left, subtraction of positive numbers is moving towards the left, and subtraction of negative numbers is moving towards the right. Multiplication by a positive number is also straightforward: if you are multiplying something by n times, you just perform the movement action n times.
But multiplication by a negative number has no meaning in the source domain of motion. You can’t “do something a negative number of times”. A new metaphor must be found, constrained by the fact that it needs to fit the fact that we’ve found 5 (-2) = −10 and that, by the law of commutation (also straightforwardly derivable from the grounding metaphors), (-2) 5 = −10.
Now:
The symmetry between positive and negative numbers motivates a straightforward
metaphor for multiplication by –n: First, do multiplication by the positive
number n and then move (or “rotate” via a mental rotation) to the
symmetrical point—the point on the other side of the line at the same distance
from the origin. This meets all the requirements imposed by all the laws. Thus,
(–2) · 5 = –10, because 2 · 5 = 10 and the symmetrical point of 10 is –10. Similarly,
(–2) · (–5) = 10, because 2 · (–5) = –10 and the symmetrical point of –10 is 10. Moreover,
(–1) · (–1) = 1, because 1 · (–1) = –1 and the symmetrical point of –1 is 1.
The process we have just described is, from a cognitive perspective, another
metaphorical blend. Given the metaphor for multiplication by positive numbers,
and given the metaphors for negative numbers and for addition, we form a
blend in which we have both positive and negative numbers, addition for both,
and multiplication for only positive numbers. To this conceptual blend we add
the new metaphor for multiplication by negative numbers, which is formulated in terms of the blend! That is, to state the new metaphor, we must use
negative numbers as point-locations to the left of the origin,
addition for positive and negative numbers in terms of movement, and
multiplication by positive numbers in terms of repeated addition a positive
number of times, which results in a point-location.
Only then can we formulate the new metaphor for multiplication by negative
numbers using the concept of moving (or rotating) to the symmetrical point-location.
So in other words, we have taken some grounding metaphors and built a new metaphor that blends elements of them, and after having constructed that new metaphor, we use the terms of that combined metaphor to define a new metaphor on top of that.
While this example was in the context of an obviously physically applicable part of maths, their argument is that all of maths is built in this way, starting from physically grounded metaphors which are then extended and linked to build increasingly abstract forms of mathematics… but all of which are still, in the end, constrained by the physical regularities they were originally based on:
The metaphors given so far are called grounding metaphors because they directly
link a domain of sensory-motor experience to a mathematical domain.
But as we shall see in the chapters to come, abstract mathematics goes beyond
direct grounding. The most basic forms of mathematics are directly grounded.
Mathematics then uses other conceptual metaphors and conceptual blends to
link one branch of mathematics to another. By means of linking metaphors,
branches of mathematics that have direct grounding are extended to branches
that have only indirect grounding. The more indirect the grounding in experience,
the more “abstract” the mathematics is. Yet ultimately the entire edifice
of mathematics does appear to have a bodily grounding, and the mechanisms
linking abstract mathematics to that experiential grounding are conceptual
metaphor and conceptual blending.
To take a step back. the discussion is about mathematical Platonism, a theory of mathematical truth which is apparently motivated by the Correspondence theory of truth. That is being rivaled by another theory, also motivated by CToT, wherein the truth-makers of mathematical statements are physical facts, not some special realm of immaterial entities. The relevance of my claim that there are unphysical mathematical truths is that is an argument against the second claim.
Lakoff and Nunez give an account of the origins and nature of mathematical thought that while firmly anti-Platonic doesn’t back a rival theory of mathematical truth, because that is not in fact their area of interest..their interest is in mathematical thinking.
That is being rivaled by another theory, also motivated by CToT, wherein the truth-makers of mathematical statements are physical facts
Who said that? Actual formal systems run on a coherence theory of truth: if the theory is consistent (and I do mean consistent according to a meta-system, so Goedel and Loeb aren’t involved right now), then it’s a theory. It may also be a totally uninteresting theory, or a very interesting theory. The truth-maker for a mathematical statement is just whether it has a model (and if you really wanted to, you could probably compile that into something about computation via the Curry-Howard Correspondence and some amount of Turing oracles). But the mere truth of a statement within a formal system is not the interesting thing about the statement!
Who said that CToT motivates mathematical Platonism, or who said that CToT is the outstanding theory of mathemtaical truth?
Actual formal systems run on a coherence theory of truth: if the theory is consistent (and I do mean consistent according to a meta-system, so Goedel and Loeb aren’t involved right now), then it’s a theory. It may also be a totally uninteresting theory, or a very interesting theory. The truth-maker for a mathematical statement is just whether it has a model (and if you really wanted to, you could probably compile that into something about computation via the Curry-Howard Correspondence and some amount of Turing oracles). But the mere truth of a statement within a formal system is not the interesting thing about the statement!
I couldn’t agree more that coherence is the best description of mathematical practice.
Insofar as logic consists in information-preserving operations, the non-physically-applicable parts of math still correspond to the real world, in that they preserve the information about the real world which was put into formulating/locating the starting formal system in the first place.
This is what makes mathematics so wondrously powerful: formality = determinism, and determinism = likelihood functions of 0 or 1. So when doing mathematics, you get whole formal systems where the theorems are always at least as true as the axioms. As long as any part of the system corresponds to the real world (and many parts of it do) and the whole system remains deterministic, then the whole system compresses information about the real world.
Insofar as logic consists in information-preserving operations, the non-physically-applicable parts of math still correspond to the real world, in that they preserve the information about the real world which was put into formulating/locating the starting formal system in the first place.
Whereas the physically inapplicable parts don’t retain real-world correspondence. Correspondence isn’ta n intrinsic, essential part of maths.
Sure, you can come up with a formal system that bears no correspondence to the real world whatsoever. Mathematicians just won’t consider it very interesting most of the time.
Sure, you can come up with a formal system that bears no correspondence to the real world whatsoever. Mathematicians just won’t consider it very interesting most of the time.
Transfinite mathematics is very interesting and currently has no correspondence to the physical world, at least not in any way that anyone knows about. And you can make the argument that even if there is a correspondence, we will never know about it, because you would have to be sure that actual infinities exist in the physical world, and that would seem pretty hard to confirm.
A lot of pure math takes the form of: “let’s take something in the real world, like ‘notion of containment in a bag’ and run off with it.” So it’s abstracting, but then it’s not about the real world anymore. There are no cardinals in the real world, but there are bags.
So it’s abstracting, but then it’s not about the real world anymore.
Yes it is. It still consists in information from the real world. The precise structure was chosen out of an infinite space of possible structures based precisely on its ability to generalize scenarios from “real life”.
Consider, for instance, real numbers and continuity. The real world is not infinitely divisible—we know this now! But at one time, when these mathematical theories were formulated, that was a working hypothesis, and in fact, people could not divide things small enough to actually find where they became discrete. So continuity, as a mathematical construct, started out trying to describe the world, and was later found to have more interesting implications even when it was also found to be physically wrong.
I think you’re engaging in deepities here. It is clearly true that all of mathematics historically descends from thoughts about the real world. It is clearly false that all of mathematics is directly about the real world. Using the same words for both claims, “mathematics is about the real world”, is the deepity.
So continuity, as a mathematical construct, started out trying to describe the world, and was later found to have more interesting implications even when it was also found to be physically wrong.
That is news to me. Physicists, even fundamental physicists, still talk about differential geometry and Hilbert spaces and so on. There are speculations about an underlying discrete structure on the Planck scale or below, but did anyone refound physics on that basis yet? Stephen Wolfram made some gestures in that direction in his magnum opus; but I read a physicist writing a review of it saying that Wolfram’s idea of explaining quantum entanglement that way was already known not to work.
I think you’re engaging in deepities here. It is clearly true that all of mathematics historically descends from thoughts about the real world. It is clearly false that all of mathematics is directly about the real world. Using the same words for both claims, “mathematics is about the real world”, is the deepity.
I think it’s at least arguable that there are plenty of cardinals in the real world (e.g., three) even though there very likely aren’t the “large cardinals” that set theorists like to speculate about.
I (1) was genuinely unsure whether you were asserting that numbers (even small positive integers) are too abstract to “live” in the real world—a reasonable assertion, I think, though I thought it probably wasn’t your position—and (2) thought it was amusing even if you weren’t.
But that’s not at all relevant The existence of unphysical maths is a robust argument against the theory that mathematical truth is true by correspondence to the physical world. The interestingness of such maths is neither here nor there.
Indeed, even though when you use a Lossy-Correspondence/Compression Theory of Truth, abstract objects become perfectly sensible as descriptions of regularities in concrete objects.
Not really, because most maths is unphysical, ie physics is picking out the physically applicable parts of maths, ie the rest has nothing to correspond to.
If I remember my Lakoff & Núñez correctly, they were arguing that even the most abstract and un-physical-seeming of maths is constructed on foundations that derive from the way we perceive the physical world.
Let me pick up the book again… ah, right. They define two kinds of conceptual metaphor:
Their argument is that for any kind of abstract mathematics, if you trace back its origin for long enough, you finally end up at some grounding and linking metaphors that have originally been derived from our understanding of physical reality.
As an example of the technique, they discuss the laws of arithmetic as having been derived from four grounding metaphors: Object Collection (if you put one and one physical objects together, you have a collection of two objects), Object Construction (physical objects are made up of smaller physical objects; used for understanding expressions like “five is made up of two plus three” or “you can factor 28 into 7 times 4″), Measuring Stick (physical distances correspond to numbers; gave birth to irrational numbers, when the Pythagorean theorem was used to prove their existence by assuming that there’s a number that corresponds to the length of the hypotenuse), and Motion Along A Path (used in the sixteenth century to invent the concept of the number line, and the notion of a number as lying between two other numbers).
Now, they argue that these grounding metaphors, each by themselves, are not sufficient to define the laws of arithmetic for negative numbers. Rather you need to combine them into a new metaphor that uses parts of each, and then define your new laws in terms of that newly-constructed metaphor.
Defining negative numbers is straightforward using these metaphors: if you have the concept of a number line, you can define negative numbers as “point-locations on the path on the side opposite the origin from positive numbers”, so e.g. −5 is the point five steps to the left of the origin point, symmetrical to +5 which is five steps to right of the origin point.
Next we can use Motion Along A Path to define addition and subtraction: adding positive numbers is moving towards the right, addition of negative numbers is moving towards the left, subtraction of positive numbers is moving towards the left, and subtraction of negative numbers is moving towards the right. Multiplication by a positive number is also straightforward: if you are multiplying something by n times, you just perform the movement action n times.
But multiplication by a negative number has no meaning in the source domain of motion. You can’t “do something a negative number of times”. A new metaphor must be found, constrained by the fact that it needs to fit the fact that we’ve found 5 (-2) = −10 and that, by the law of commutation (also straightforwardly derivable from the grounding metaphors), (-2) 5 = −10.
Now:
So in other words, we have taken some grounding metaphors and built a new metaphor that blends elements of them, and after having constructed that new metaphor, we use the terms of that combined metaphor to define a new metaphor on top of that.
While this example was in the context of an obviously physically applicable part of maths, their argument is that all of maths is built in this way, starting from physically grounded metaphors which are then extended and linked to build increasingly abstract forms of mathematics… but all of which are still, in the end, constrained by the physical regularities they were originally based on:
To take a step back. the discussion is about mathematical Platonism, a theory of mathematical truth which is apparently motivated by the Correspondence theory of truth. That is being rivaled by another theory, also motivated by CToT, wherein the truth-makers of mathematical statements are physical facts, not some special realm of immaterial entities. The relevance of my claim that there are unphysical mathematical truths is that is an argument against the second claim.
Lakoff and Nunez give an account of the origins and nature of mathematical thought that while firmly anti-Platonic doesn’t back a rival theory of mathematical truth, because that is not in fact their area of interest..their interest is in mathematical thinking.
Who said that? Actual formal systems run on a coherence theory of truth: if the theory is consistent (and I do mean consistent according to a meta-system, so Goedel and Loeb aren’t involved right now), then it’s a theory. It may also be a totally uninteresting theory, or a very interesting theory. The truth-maker for a mathematical statement is just whether it has a model (and if you really wanted to, you could probably compile that into something about computation via the Curry-Howard Correspondence and some amount of Turing oracles). But the mere truth of a statement within a formal system is not the interesting thing about the statement!
Who said that CToT motivates mathematical Platonism, or who said that CToT is the outstanding theory of mathemtaical truth?
I couldn’t agree more that coherence is the best description of mathematical practice.
This one.
Or rather, who claimed that the truth-makers of mathematical statements are physical facts?
Insofar as logic consists in information-preserving operations, the non-physically-applicable parts of math still correspond to the real world, in that they preserve the information about the real world which was put into formulating/locating the starting formal system in the first place.
This is what makes mathematics so wondrously powerful: formality = determinism, and determinism = likelihood functions of 0 or 1. So when doing mathematics, you get whole formal systems where the theorems are always at least as true as the axioms. As long as any part of the system corresponds to the real world (and many parts of it do) and the whole system remains deterministic, then the whole system compresses information about the real world.
Whereas the physically inapplicable parts don’t retain real-world correspondence. Correspondence isn’ta n intrinsic, essential part of maths.
Sure, you can come up with a formal system that bears no correspondence to the real world whatsoever. Mathematicians just won’t consider it very interesting most of the time.
They call it “pure mathematics”.
Transfinite mathematics is very interesting and currently has no correspondence to the physical world, at least not in any way that anyone knows about. And you can make the argument that even if there is a correspondence, we will never know about it, because you would have to be sure that actual infinities exist in the physical world, and that would seem pretty hard to confirm.
What do cardinals correspond to?
I suppose it’s about the language, not about the model. Still.
Screw it. I’ll just go do a PhD thesis on how abstraction works, and then anyone who wants to actually understand can read that.
A lot of pure math takes the form of: “let’s take something in the real world, like ‘notion of containment in a bag’ and run off with it.” So it’s abstracting, but then it’s not about the real world anymore. There are no cardinals in the real world, but there are bags.
Yes it is. It still consists in information from the real world. The precise structure was chosen out of an infinite space of possible structures based precisely on its ability to generalize scenarios from “real life”.
Consider, for instance, real numbers and continuity. The real world is not infinitely divisible—we know this now! But at one time, when these mathematical theories were formulated, that was a working hypothesis, and in fact, people could not divide things small enough to actually find where they became discrete. So continuity, as a mathematical construct, started out trying to describe the world, and was later found to have more interesting implications even when it was also found to be physically wrong.
Ok, so you would say inaccessible cardinals, versions of the continuum hypothesis etc. are “about the real world.”
At this point, I am bowing out of this conversation as we aren’t using words in the same way.
I think you’re engaging in deepities here. It is clearly true that all of mathematics historically descends from thoughts about the real world. It is clearly false that all of mathematics is directly about the real world. Using the same words for both claims, “mathematics is about the real world”, is the deepity.
That is news to me. Physicists, even fundamental physicists, still talk about differential geometry and Hilbert spaces and so on. There are speculations about an underlying discrete structure on the Planck scale or below, but did anyone refound physics on that basis yet? Stephen Wolfram made some gestures in that direction in his magnum opus; but I read a physicist writing a review of it saying that Wolfram’s idea of explaining quantum entanglement that way was already known not to work.
I’m actually only using it for the former.
You may be thinking of Scott Aaronson’s review of “A new kind of science”.
I think it’s at least arguable that there are plenty of cardinals in the real world (e.g., three) even though there very likely aren’t the “large cardinals” that set theorists like to speculate about.
(Of course there are also cardinals and cardinals.)
What was the point of writing that? Do you think I was talking about “3”?
I (1) was genuinely unsure whether you were asserting that numbers (even small positive integers) are too abstract to “live” in the real world—a reasonable assertion, I think, though I thought it probably wasn’t your position—and (2) thought it was amusing even if you weren’t.
But that’s not at all relevant The existence of unphysical maths is a robust argument against the theory that mathematical truth is true by correspondence to the physical world. The interestingness of such maths is neither here nor there.