Arstechnica (hey, they’ve had some back-to-back articles I want to comment on!) has posted an article discussing a panel talking about the Limits of Understanding. Particularly, the panel seemed to focus on apparent deficiencies in math – notably, the practical intractability of biology to mathematical modeling and Godel’s Incompleteness Theorem.

Now, the proper modeling of biology may simple be a problem of time…sometime, a mathematical model may be devised of sufficient complexity and rigor that it is considered an adequate analysis of the underlying systems leading to observable reality. What exercises the mathematicians more is whether an “adequate model” is a true model, that is.

The statement which interests me the most, though, is the final two sentences, uttered by the author of the article himself:

If math turns out to be just a tool (and a tool with some substantial limits), that may disappoint mathematicians, but it won’t necessarily slow down our ability to understand and model the natural world. This may be my background as a scientist talking, but that seems like the most important consideration, and I’m willing to live with a community of disappointed mathematicians in order to get there.

This demonstrates an infuriating philosophical stance. A model does two things: it relates observations to one another, expressing a causal relationship chain between them, and elucidates the underlying real factors which produce these observations. There exist two parts to the natural world: the observable, particular parts and the invisible relations underlying them. Two rocks are observable, the force holding them to the ground, or the force value we say they exhibit as a property is not. We infer force, based on changes over time. Force isn’t so much real as a very successful description of an inferred relationship. Mass is similar: it’s a relationship one objects exhibits vis a vis all other possible objects. Importantly, mass was very successfully describes as a constant property of a given set of material. However, mass can equally well be described as a property of curved space surrounding an object (if we’re talking about the source of the “attractive” force) or as a measure of the type of curvature produced by a massive object.

The statement by the author above exhibits a certain lack of philosophical follow through. A model which successfully relates observed experience to one another implicitly makes statements about the invisible relationships observations have. We cannot “understand and model” the natural world correctly unless we also correctly model those invisible relationships. Unfortunately, those relationships are invisible; the best we can do is infer their nature. If we have inferred incorrectly, then we are wrong, and our understanding is flawed. We DON’T understand. Since these relationships are invisible, this is somewhat untestable, so we won’t know. The hope is that, by using rigorously consistent systems, systems which exhibit the same properties as the fundamental relational properties of existence, we can leave behind some of those inferential worries.

What is irritating the mathematicians is that Godel seems to have annihilated that possibility. What really frustrates them is that if Godel truly killed the possibility of a formal system being provable, or that it is composed of two distinct subsets of theorems: those which are provable and those which are not, then do the relationships underpinning the universe dividable into a set of provable relations? That is, do all the relationships of the universe derive from a rational order. Importantly, Godel showed that certain theorems of a given system cannot follow from an entire set of other theorems. It’s like saying a relationship could be true, or false, and it bears no relation to other theorems. That’s fine, you might say, we can just look and see, right?

That’s part of the mathematicians problem, though: you can’t. We’re speaking here of the fundamental relations of the universe, not the observable particulars. Physics and biology aren’t much use if they can’t predict what’s going to happen, based on other things it knows. Godel showed that, if you have a formal system of rules that is good enough to describes math, then there are going to be statements in that system YOU CAN’T PROVE. Which means you can’t predict. If someone asks, you have to say “I don’t know”. Hofstadter shows (in *Godel, Escher, and Bach*) that there exists an infinite set of such statements.

If we take the positivist approach of simply using math as models where appropriate, we’ve thrown up our hands and admitted: we can’t actually describe the universe. We can describe bits of it, but we can’t relate it together. Further, because we cannot assert that the universe is describable by fundamental, consistent laws – because there cannot exist any such systems, including the universe – we can’t actually say the models we use are true, correct descriptions of how the universe works. That’s not decidable. All we can say is they work. There demonstrably exists at least one other formal system which can accurately describe the observable phenomena a model describes, and it is impossible to decide between them. In fact, it’s not even possible to say the universe is rational, because that, too, is undecidable.

If the purpose of the University is the search for truth, then it becomes problematic for the University if it can be rationally demonstrated that Truth is unreachable. That means its purpose is finished: it has found the one Truth: there is no Truth. Everything else is mental masturbation.

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## New Keynsian Tinkerbell and an Explanation for How She Flies

Posted in Commentary, Economics on June 7, 2010| Leave a Comment »

Nick Rowe has an excellent post discussing what he calls a “Tinkerbell” (pulled from a Krugman criticism) – a poof of fairy dust that makes New Keynsian models work. This is coming from a New Keynsian, and is something of an introspection on their models and how they work.

His example case is real interest rates, and I’ll just let him lay out the base of the model behind the problem:

According to the Keynsian model (I’m going to use the more standard US terminology of “Keynsian” and “Post-Keynsian”), real interest rate reductions increase current demand. This occurs because real interest rate reductions disincentivize savings and thus increases demand. The Post-Keynsian model put forward by Rowe adds a nuance to this, stating that a reduction of the real interest rate increases demand relative to future demand. In effect, Post-Keynsians (or this Post-Keynsian model) argue that the real interest rate impacts the ratio between current demand and future demand, rather than strictly impacting demand. The Post-Keynsian model offers a nuanced view, of which the Keynsian model is a special case. However, Rowe points out it’s impossible to determine what will occur upon reducing real rates: current demand could rise, and future demand could fall…or current demand could fall and future demand could fall…it just might fall more. The model provides no way to make that determination. The only thing, according to Rowe, a Post-Keynsian might offer up is the belief that the economy will reah full employment at some future time, so it all works out. But they can’t demonstrate that…it’s a belief in magic. A Tinkerbell.

He ends by suggesting that perhaps monetary policy is about the supply and demand for money. I would say it is rather about the supply and demand of cash flows, where money itself is (generally) simply a discontinuous measure of cash flows. Since a cash flow can be described as an interest rate, then we can say that interest rates ARE the focus of monetary policy. However, I am using the term much more broadly in that sentence than is generally meant. Rowe is right: the perspective of monetary policy perhaps should be broadened.

I would also like to offer a few steps along the resolution of the Tinkerbell issue described above. This isn’t intended to suggest all the Tinkerbells of Post-Keynsian thought can be cleared away, but this one, I think, can. What’s important is to think about the transmission effects which lead to the observation that reducing real interest rates increases the ratio of current to future demand. When interest rates are reduced, this impacts savers and borrowers (or purchasers and sellers of cash flows). Savers are less inclined to save because they’ll get less; they’re going to tend to want to try and find more bang for their buck. Borrowers will be more inclined to borrow, because the lower interest rate represents a reduced negative cash flow. Now, tacitly, this implies that they will increase demand now. Further, lower interest rates imply reduced future cash flows, for similar reasons, which leads to lower future demand relative to current demand.

However, we run into a couple points where interest rate changes do not alter present demand. For borrowers, there exist points where current cash flows are negative – that is, they can’t cover their existing costs. While additional borrowing can stave this off, that will produce an inflection point where the cost of interest payments (the negative cash flow incurred from borrowing) grow to the point where they dominate the other cash flows. Income flows + borrowing cease to be enough to cover both interest payments and living expenses. In these cases, or in cases near to them, reduced real interest rates will not induce increased borrowing.

Additionally, savers may hit a point where, regardless of interest rates, no other possible use of money seems to offer the same benefit to their cash flows. The easiest example of this case is someone who feels high uncertainty regarding the future. While they may be put off by the reduced return on cash flows they can purchase, if they are concerned that their other cash flows may be put in jeopardy in the future (by, say, losing their job), they will be inclined to save regardless. A certain cash flow is an offset for uncertain cash flows.

Now, I am less sure about future effects. In fact, I am not convinced that the ratio between current and future demand is always increased by a reduction in real interest rates. The assumption, obviously, is that aggregate cash flows must have been reduced by the interest rate reduction. This feels…incorrect to me. I guess you have to assume that cash flows always tend to increase? What happens when cash flows decline? Because cash flows can decline. A credit is offered on future ability to pay. If I take out a loan, and then lose my job and can’t pay the loan, the bank is screwed. It doesn’t matter what the bank does, they’re unlikely to be able to recover the full cash flow. Suddenly, cash flows have vanished. Obviously this can happen in aggregate: we just went through such an event.

However, we also conceive of cash flow increases REGARDLESS of real interest rates. If those can be catalyzed by real interest rate tweaks, then one can induce a situation where a reduction in real interest rates actually increases future demand.

Then again “future demand” is somewhat poorly defined now.

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