# Math Help - theories without countable models versus measurement

1. ## theories without countable models versus measurement

Of course, one has mathematics which is pursued for its own sake. However, suppose one wished to justify the connection of a theory to the physical world (beyond the trivial bit about the physical brain that thinks it up). The problem is that measurements are all rational numbers, but we posit theories with uncountable models. (I am thinking of the class of measurements as the universe of my model.) Many theories which have uncountable models can be justified by the Löwenheim-Skolem Theorem, whereby any first-order theory with an uncountable model also possesses a countable model, so we can say that in using the uncountable interpretation we are just making a shortcut, but in reality we are describing a countable model. However, there are second-order theories which are not reducible to first-order, and where the L-S Theorem fails. Hence the previous justification also fails. Perhaps because of this failure most papers concentrate on first-order theories, but nonetheless second-order theories prove useful in describing certain phenomena. But apart from the justification "it's useful, so use it", can one justify using such a theory which does not correspond to a model with the universe being restricted to the class of possible measurements?

Of course, one has mathematics which is pursued for its own sake. However, suppose one wished to justify the connection of a theory to the physical world (beyond the trivial bit about the physical brain that thinks it up). The problem is that measurements are all rational numbers, but we posit theories with uncountable models. (I am thinking of the class of measurements as the universe of my model.) Many theories which have uncountable models can be justified by the Löwenheim-Skolem Theorem, whereby any first-order theory with an uncountable model also possesses a countable model, so we can say that in using the uncountable interpretation we are just making a shortcut, but in reality we are describing a countable model. However, there are second-order theories which are not reducible to first-order, and where the L-S Theorem fails. Hence the previous justification also fails. Perhaps because of this failure most papers concentrate on first-order theories, but nonetheless second-order theories prove useful in describing certain phenomena. But apart from the justification "it's useful, so use it", can one justify using such a theory which does not correspond to a model with the universe being restricted to the class of possible measurements?
If it predicts accurately, then I'd say use it until something better comes along (such as Einstein's Theory of Relativity which more accurately predicts than Isaac Newton's).

3. Thanks, wonderboy1953, but I would say that you have missed the point of the post. I put in
apart form the justification "it's useful, so use it"
, and your answer was essentially "it's useful, so use it". Of course one uses it, but I see a problem similar to the one posed by the paradoxes at the beginning of the 20th century: then, too, the day-to-day mathematics was not affected by the crisis in Foundations; the mathematics used was justifiable, but whereas the usage was correct, the justification turned out to be wrong, so one had to look for the correct justification. To give a more basic example: if a school student gets the right answer with a faulty method, he doesn't get any points. ("What is the length of the hypotenuse of a right triangle with legs 13 and 84."...."85".... Good. How did you get it so fast?"...." Simple. I saw that in the 3-4-5 triangle, the 5-12-13 triangle, the 7-24-25 triangle, the 9-40-41 triangle, and the 11-60-61 triangle, the hypotenuse was one plus the longest leg. This is apparently also such a case." Zero marks.)

Thanks, wonderboy1953, but I would say that you have missed the point of the post. I put in , and your answer was essentially "it's useful, so use it". Of course one uses it, but I see a problem similar to the one posed by the paradoxes at the beginning of the 20th century: then, too, the day-to-day mathematics was not affected by the crisis in Foundations; the mathematics used was justifiable, but whereas the usage was correct, the justification turned out to be wrong, so one had to look for the correct justification. To give a more basic example: if a school student gets the right answer with a faulty method, he doesn't get any points. ("What is the length of the hypotenuse of a right triangle with legs 13 and 84."...."85".... Good. How did you get it so fast?"...." Simple. I saw that in the 3-4-5 triangle, the 5-12-13 triangle, the 7-24-25 triangle, the 9-40-41 triangle, and the 11-60-61 triangle, the hypotenuse was one plus the longest leg. This is apparently also such a case." Zero marks.)
Apart from the justification "it's useful, so use it" I don't really see any justification for using mathematics at all. The current setup we have is useful, so it's important to do things correctly within that setup.

You can gloss over the "all measurements are rationals" thing by just accepting that instead of observing $X = x$ you are actually observing $X \in A$ and using the former as a highly useful approximation to the latter.

5. Originally Posted by theodds
Apart from the justification "it's useful, so use it" I don't really see any justification for using mathematics at all. The current setup we have is useful, so it's important to do things correctly within that setup..
A lot of people study mathematics for its own sake. In fact, most mathematics will never be used in any practical way. Be that as it may, the correctness can be a problem if it appears to give correct answers now, but could eventually turn nasty due to some hidden contradiction. The contradictions in applied mathematics can be external as well as internal contradictions: it should agree with the world. There is a lot of finite mathematics that is used; the question is whether that which cannot be expressed in these terms can agree with the world.

Originally Posted by theodds
You can gloss over the "all measurements are rationals" thing by just accepting that instead of observing $X = x$ you are actually observing $X \in A$ and using the former as a highly useful approximation to the latter.
That is accepting a view of reality which is not in line with most physicists today, whereby reality and measurement are the same thing, and all the rest is simply a bunch of shortcuts for long ways which contain only physically meaningful steps, but which are either too tedious or not yet discovered. But the question remains whether those shortcuts for which there is no corresponding way (using only physically meaningful steps) are hiding an eventual clash, just as Zeno's paradoxes hid clashes that became important only much later.

A lot of people study mathematics for its own sake. In fact, most mathematics will never be used in any practical way. Be that as it may, the correctness can be a problem if it appears to give correct answers now, but could eventually turn nasty due to some hidden contradiction. The contradictions in applied mathematics can be external as well as internal contradictions: it should agree with the world. There is a lot of finite mathematics that is used; the question is whether that which cannot be expressed in these terms can agree with the world.
Sure, I don't disagree with this, although to be honest it isn't exactly clear to me in what sense you expect the mathematical setup to conform to reality. Expecting mathematics and reality to conform to each other in some exact sense is, IMO, a pipe dream.

That is accepting a view of reality which is not in line with most physicists today, whereby reality and measurement are the same thing, and all the rest is simply a bunch of shortcuts for long ways which contain only physically meaningful steps, but which are either too tedious or not yet discovered. But the question remains whether those shortcuts for which there is no corresponding way (using only physically meaningful steps) are hiding an eventual clash, just as Zeno's paradoxes hid clashes that became important only much later.
I'm not really a physicist, but my gut says this is stretching the actual mainstream view. $X \in A$ is a measurement, and I don't think any physicist ever would think they observed something like $X = x$ by comparison. I'm not accepting any view of reality at all; merely stating the fact that human beings don't make precise measurements, and that only pretend they do to facilitate calculations. The fact that it is physically impossible for us to measure X = x is a problem with us, not a problem with our models (i.e. if there is a problem with the model it isn't because of this, it is something else).

7. Originally Posted by theodds
Sure, I don't disagree with this, although to be honest it isn't exactly clear to me in what sense you expect the mathematical setup to conform to reality.

The mathematical model should provide values towards which the expected values (in the statistical sense) of physical measurements converge. For example, I don't disallow pi just because we do not ever measure a circle to infinite values, since pi provides values around which the measured values are distributed.

Originally Posted by theodds
$X \in A$ is a measurement, and I don't think any physicist ever would think they observed something like $X = x$ by comparison.
Before I can comment on this, I think I had better ask you to make precise what you mean by measuring $X \in A$ and $X = x$. It seems to me that you are either saying what I just wrote above, or that you are using A and x in some Platonic sense, but this could be presumptuous of me, so please indicate what you mean. Thanks.

8. ## Re: theories without countable models versus measurement

we believe (this is pretty much an article of faith, or inductive logic) that the universe is consistent. therefore, we look for models, with consistency being one of the chief requirements for "modelling reality". in this regard, mathematics has served well.

although something like geometry was developed to measure the world we live in (hence its name), we find ourselves in the curious position of not actually knowing the geometry of the universe we live in (despite some assiduous attempts to discern it). Minkowski space-time (and its low-velocity Euclidean approximation) seems to be a "good fit", but has its detractors.

as an aside, i would like to point out that the countability of our "measurements" is more a feature of how we measure, than of what we are measuring. there is an infinite recursion problem with defining a unit of measurement, which forces us to choose one arbitrarily (what ruler do you measure the accuracy of the rulers with?). having chosen a unit, we are limited to constructible numbers (constructible here meaning a somewhat larger set than the classical "constructible" numbers, more akin to the "calculable" or "computable" numbers).

in other words, just because WE are finite, and thus are only capable of producing finitary measurements, does not mean that the universe is finite (although it might be). we are only capable of a level of resolution; ultimately, while we may speculate that some extension of our scale of resolution may actually be the case, it is impossible for us to know (but that apparently does not deter us from trying).

9. ## Re: theories without countable models versus measurement

we believe (this is pretty much an article of faith, or inductive logic) that the universe is consistent.
If, by the universe, you mean the physical universe, then I do not think it is an article of faith or inductive logic, but rather a tautology: it cannot be otherwise. Can you give me a model in which the physical universe is inconsistent? In Physics, whenever you have what appears to be A & ~A, upon investigation we have (A,B)&(~A,C): for example, time and space. Even quantum physics is not contradictory, just counter-intuitive.
the countability of our "measurements" is more a feature of how we measure, than of what we are measuring.
Not according to the present view of Physics that reality is the same thing as measurement. This view was introduced with Relativity, and taken further by quantum physics. "Objective reality" independent of measurement belongs to religion, not to Physics, even though in some contexts it is convenient, because it falls in line with our hard-wired intuition (which have been nicely termed "necessary falsehoods"), to talk as if there is one.
...does not mean that the universe is finite (although it might be)
The possible infinite universe is the convenient but not necessarily minimal model which would satisfy all the physical laws. That it is not minimal is seen by the the Löwenheim-Skolem theorem , since our theory, couched in the countable language of measurement, can also be satisfied by a countable one. Going further, given the finite nature of measurement, such a countable model could be taken to be an unbounded collection of finite models. This is true whether you are referring to the infinitude of elements or the infinitude (large or small) of extension.

10. ## Re: theories without countable models versus measurement

If, by the universe, you mean the physical universe, then I do not think it is an article of faith or inductive logic, but rather a tautology: it cannot be otherwise. Can you give me a model in which the physical universe is inconsistent? In Physics, whenever you have what appears to be A & ~A, upon investigation we have (A,B)&(~A,C): for example, time and space. Even quantum physics is not contradictory, just counter-intuitive.
imagine that the universe was completely determined by a set of 6 differential equations. imagine that this system of differential equations had 2 subtly differing, but equally valid solutions, and that the differences could only be noticed on (small) physical scales beyond our current abilities to observe, or on very long time-scales. suppose further, that the state space of our universe oscillated aperiodically between the two solutions, and that one system lead to a dynamic equilibirium, and the other system lead to total annihilation. so, at some point in time, we suddenly switch to a universe about to be destroyed, in total violation of all our "known" theories. why can't this be the case?

your example is inductive: what is actually the case is: "every time we have investigated what appears to be A&~A, we have found (A,B)&(~A,C)". there is no reason to believe that this will always be the case, it just appears reasonable to make that assumption. physicsts sometimes start to act as if what they theorize is true....this is dangerous ground...the BEST we can say is that experience bears out our predictions...so far. we have a high degree of statistical confidence in our models of the worlds...but they are NOT absolute.

Not according to the present view of Physics that reality is the same thing as measurement. This view was introduced with Relativity, and taken further by quantum physics. "Objective reality" independent of measurement belongs to religion, not to Physics, even though in some contexts it is convenient, because it falls in line with our hard-wired intuition (which have been nicely termed "necessary falsehoods"), to talk as if there is one.
that is a view, and does not mean it is the correct view. it may be the case that we can't actually "know how things are" and will have to content ourselves with "how we can measure how things are". that does not mean that a belief in an underlying reality is automatically invalid. the underlying assumptions of physics are still beliefs about the world, the experimental validation of them notwithstanding. to adopt these underlying assumptions as "true" and dismiss other points of view as "religious" is both narrow-minded, and incorrect. i am sure there are plenty of objectivists who still believe in science.

as far as i am aware (and this could be out-of-date, if so, my apologies), the current measurement of time is based upon a standard based on the transition states of cesium-133. obviously we'd have a more consistent unit of measurement if we based this on say, some regular feature of a subatomic particle (just how sure are we of the regularity of all cesium-133 atoms, anyway?). according to your view, physicists are "expanding reality" with every new discovery. this is rather close to solipsism. if reality is just measurement, what is it that we are measuring?

The possible infinite universe is the convenient but not necessarily minimal model which would satisfy all the physical laws. That it is not minimal is seen by the the Löwenheim-Skolem theorem , since our theory, couched in the countable language of measurement, can also be satisfied by a countable one. Going further, given the finite nature of measurement, such a countable model could be taken to be an unbounded collection of finite models. This is true whether you are referring to the infinitude of elements or the infinitude (large or small) of extension.
if i understand you correctly, you are really asking: why do we need second-order theories? i don't think we have the answer to that. certainly, some second-order theories are useful, but that's only a "stop-gap" answer. perhaps the question is unanswerable, a sort of meta-mathematical incompleteness theorem. the whole of the situation is rather complex: on the one hand, we want to be able to make meaningful, consistent statements which yield insight and knowledge of the (whether "measureable" or "actual") world we live in, on the other, we are tangled in a mire of what does "meaningful" "mean", and how "consistent" is "consistency".

how "categorical" do we wish to allow our logical systems to become, to what ends, and at what cost? at one end of the spectrum you have those arguing for a minimal set of assumptions, and a minimal set of allowable rules, yielding forms of expression that are nearly indecipherable, i defy you to explain to a lay-person the meaning of a first-order proof in a first-order formal language of the countability of the theory of real-closed fields (in detail, the reader's digest version doesn't count). at the other end, we have people arguing for a flexible and open-ended system with a plethora of rules and structures, that allow for concise statements of far-ranging results (example: the first isomorphism theorem). here, communication isn't the problem, it's the shaky foundations underpinning them.

so who's right? and what is our yardstick for deciding? opinions differ.

11. ## Re: theories without countable models versus measurement

imagine that the universe was completely determined by a set of 6 differential equations. imagine that this system of differential equations had 2 subtly differing, but equally valid solutions, and that the differences could only be noticed on (small) physical scales beyond our current abilities to observe, or on very long time-scales. suppose further, that the state space of our universe oscillated aperiodically between the two solutions, and that one system lead to a dynamic equilibirium, and the other system lead to total annihilation. so, at some point in time, we suddenly switch to a universe about to be destroyed, in total violation of all our "known" theories. why can't this be the case?
Perfectly valid possibility. Our theories of the universe can be inconsistent. When I said that the universe itself cannot be inconsistent, I was taking the stance (more about this below) of an "objective reality", of which we only have finite measurements, and so it cannot be known whether this objective reality is finite or infinite. The term"contradiction" is applicable to our theories of the universe, but simply not applicable to a state of nature. This is inherent in the very definition of a contradiction as one that cannot be satisfied by any model. That means that nature cannot satisfy a contradiction. Anyway, your example does not even indicate an inconsistent theory: the state which leads to total annihilation could be a perfectly consistent system.

your example is inductive: what is actually the case is: "every time we have investigated what appears to be A&~A, we have found (A,B)&(~A,C)". there is no reason to believe that this will always be the case,
Given that a contradiction cannot be satisfied in nature (see above), then we can say that there is a structure in which the phenomenon exists, and if this phenomenon both exists and doesn't exist, it is two different phenomena. Hence, what one thinks of as A and ~A must in reality be B and C. This can always be done for an incomplete mathematical theory. Given that time makes all theories for physics incomplete, this will be able to be done as long as there are physicists to do it.

Physicists sometimes start to act as if what they theorize is true....this is dangerous ground...the BEST we can say is that experience bears out our predictions...so far. we have a high degree of statistical confidence in our models of the worlds...but they are NOT absolute.
In absolute agreement.
it may be the case that we can't actually "know how things are" and will have to content ourselves with "how we can measure how things are".
This gives rise to those who equate the two. I am not defending this position, though, only stating that it is a defensible one, and hence that its opposite cannot be used as a justification for the way we do things.
that does not mean that a belief in an underlying reality is automatically invalid. the underlying assumptions of physics are still beliefs about the world, the experimental validation of them notwithstanding. to adopt these underlying assumptions as "true" and dismiss other points of view as "religious" is both narrow-minded, and incorrect. i am sure there are plenty of objectivists who still believe in science.
It was never a contention that this belief is invalid. The belief that there is an underlying reality beyond measurement is of course permitted for a physicist.... in fact, probably most physicists are philosophical materialists (objectivists, as you term them). As far as truth, the belief cannot be labeled either absolutely true or absolutely false, since there are models which satisfy both this belief and its negation. But the point is that there is no way to measure this underlying reality, by definition, and hence non-falsifiable, and hence not Physics. The basic argument of those who think that this underlying reality can be included in Physics (Gödel arguing against Hume) is that one can create a theory that expands the domain of the theories which rely only on measurement in analogy to creating the Gamma function via analytic continuation. It is an intriguing argument, but the analogy fails when there is no unique continuation. My labeling a belief as religion is not meant to be offensive; I am just separating Physics from religion. I have nothing against religious Physicists, as long as the religion does not contradict the physics.

we'd have a more consistent unit of measurement if we based this on say, some regular feature of a subatomic particle (just how sure are we of the regularity of all cesium-133 atoms, anyway?)
You are right that there is an infinite regression in all systems of measurement. The best is to base everything on invariants (speed of light, rest mass of an electron, that sort of thing), but even these are measured.
according to your view, physicists are "expanding reality" with every new discovery.
According to one's definition of reality, this is a possible position.
this is rather close to solipsism.
Close, but not the same thing. Given both the incompleteness of complex mathematical systems and the indeterminism of physical systems, a solipsist cannot predict the future, and so cannot be producing the future.
if reality is just measurement, what is it that we are measuring?
Then measuring is the same thing as experiencing, and one seems to have no problem in saying that we are experiencing reality, allowing for the possibility that it is a redundant phrase. Anyway, I would prefer to avoid the ambiguous bald word "reality" and stick to "reality accessible to Physics."
why do we need second-order theories?..... some second-order theories are useful, but that's only a "stop-gap" answer.
All of Physics is, in this sense, stop-gap. As Hawking points out, we use what works. Greene adds: what works elegantly. This is perhaps the only justifiable position. My original question was whether one can find any other reason for using second-order theories.
we are tangled in a mire of what does "meaningful" "mean",
For a closed system, I am quite satisfied with the model-theoretic definition: a statement is meaningful if it is satisfied under a model. But time makes reality an open-ended system, so if we describe the scientific method as a sort of Kripke structure, then it makes sense to say that something is meaningful one day and meaningless another.

and how "consistent" is "consistency".
I don't think there is any question as to what consistency is, but rather, given systems which are inherently incomplete, what we will accept. In Physics there are two issues: internal and external consistency. In the former, the same criteria as for mathematics is accepted, in accepting consistency relative to arithmetic. In the latter, acceptance is always based on a confidence level, whereby either a counter-example, a lack of examples, or a lack of deduction from prevailing theory minimizes the confidence level.

how "categorical" do we wish to allow our logical systems to become, to what ends, and at what cost?
Define "categorical", please. The key, I think, is that a logical system should try to avoid contradiction, and after that, all is allowed. Then the physicists use those systems which, as mentioned above, work. (Elegantly, if there is a choice.)

at one end of the spectrum you have those arguing for a minimal set of assumptions, and a minimal set of allowable rules, yielding forms of expression that are nearly indecipherable,
Not quite. What you can have is a minimal set of assumptions (William of Okham, are you listening?) but with a conservative extension which will allow for manageable sets of expressions.
so who's right?
As in certain well-known political conflicts (kept unnamed so as not to irritate the moderator), probably no one. :-)
and what is our yardstick for deciding? opinions differ.
Indeed. So, back to my original question: yes, uncountable models grease the cogs of mathematics and hence of Physics. But is there any reason to think that a minimal model, even if one is never able to formulate it (which seems likely), has to have an uncountable number of elements in its universe?

PS: above perhaps I should have used the word "metaphysics" instead of "religion". Since a lot of religions don't have a divinity and theology also does a lot of reasoning, I'm not sure of the difference between metaphysics and religion, but I guess it is there somewhere.

12. ## Re: theories without countable models versus measurement

i would argue that philosophy is the proper domain of such speculations about "the nature of reality". historically, there has been considerable overlap between philosophy and religion, as many philosophers strove to reconcile rationality and faith,or reject faith on the basis of rationality.

it could be, (i probably won't live long enough to see the outcome) that the constructivists were right, and that the current second-order theory of the real numbers isn't "accurate" as a model for scientific investigation of the world. it would be interesting to see how this affects such fields as topology and differential geometry.

as far as "categorical" goes: i chose that word for its dual connotations (yes, sometimes i out-do myself in the double-entendre department), in the ordinary sense of english, and also in the sense of category theory. that is to say, an all-encompassing view of mathematics, rather than a monolithic one. while i can appreciate the hard work many people have made into investigating the logical foundations of mathematics, i myself am inclined to take the view that "local consistency" of mathematics is all that is required; but then again, i am by and large unconcerned whether mathematics reflects actual "truth" or not. i see it more as an art form of sorts, if reality wants to play along, well...ok.

as far as my example goes, well it would be rather odd, if we suddenly blinked out of existence, and then, and some point, suddenly re-appeared, with no apparent explanation. but i agree one could develop a consistent system where even this "inconsistency" is possible. but such statements as "nature cannot satisfy a contradiction" sound unprovable to me. i feel it is more accurate to say "our models of nature cannot satisfy a contradiction" which perhaps reveals more about us, then it does about nature. in fact, i would go even further, and postulate that it may even be the case that there is no rhyme and reason to existence, and that what we perceive as such may be an artifact of the way our brains store information. i suspect that this is false, but i'm not sure how we would test it.

and i'm quite confortable with accepting different viewpoints of "how things are" at different times. i suppose i feel like what is going on is "higher dimensional" than the context we can put it in, so that what we have before us, is a "projection" of reality (in the sense of linear algebra, say), so just like different cross-sections of a 3 dimensional solid can have widely varying shapes, our "grasp" of what is occuring can take on different forms.

in the end, i suppose i agree with you: we can't totally justify our means on their own terms, and so we appeal to their utility. part of me hopes it stays that way.