The Isomorphism is not the Terrain

—Draft—

We do not often think of mathematical models as tools so much as the embodiment of a theory. We take formulae such as f=ma to be the theory, not a tool for understanding the theory. Yet mathematical formalisms are chosen for their user-friendliness —their tractability and solvability— just as much as for their correctness, which suggests that theories which rely on them may be calculable yet less than true.

Shannon's Information Theory, for instance, provided a tractable solution to the problem of what is the very best rate of message transfer we can achieve over a wire, and has been so widely re-used one might think that all the world's an abstract communication channel.

By contrast, Sundman's 1912 infinite series solution to the three-body problem languishes in obscurity not because it is wrong, but because it is incalculable. Instead, dozens and hundreds of models of special cases of the 3-body problem have been published since then, each having the virtue of user-friendliness: you can actually use them to calculate something.

With a theory such as Shannon's, usability leads to enthusiastic adoption and enthusiastic adoption leads to a sense of well-being and belief we can solve all the things with this one simple trick.

What it does not lead to, is a careful enumeration of assumptions made in the adoption of a model.

This is fine for the engineer who just wants to calculate and get a result. It is less fine when the scientist enthusiastically declares that their mathematical model solves all the philosophical problems too.

For the philosophical problems are rarely captured in the mathematical model. They are more usually either captured or carefully avoided in some well-worded definitions and assumptions in the preamble and the prerequisites of the model.

For instance, Shannon defines a measure of information: “If the number of messages in the set [of all possible message you might want to send] is finite then this number or any monotonic function of this number can be regarded as a measure of the information produced …”

The monotonic function chosen is the logarithm and leads to a definition of Shannon Information as the negative log probability of choosing that message out of all possible messages. This definition has been extremely usable because it is tractable: you can compute with it.

Sadly it has only a tangential relationship to the meaning of “information”, as Shannon noted: “Frequently messages have meaning … These semantic aspects of communication are irrelevant to the engineering problem.”

Imagine for instance, you drive from Durham to Barnard's Castle and lost at a crossroads, ask a farmer for directions. He draws his fair 6-sided die from a pocket, tosses it, and reports, “It's a 6.”

Per the mathematical model, the farmer is rather informative. He has conveyed 2.6 shannons of information. Had he instead said, “Ah yes, for Barnard's Castle you want to turn left here” that would have conveyed barely 2 shannons of information (less, if “do a U-turn” was not on the table) and so been less informative.

The model is not reality. Information in the model is not information in the real world. Rather, information in the model is something that is isomorphic to something that is quite close to information in the real world.

The model definition of information — negative log likelihood — is certainly good and very usable for a definition of the maximum possible information rate achievable over a communication channel. It has also turned out to be very … usable … in related applications. Because the idea of an abstract communication channel can be used for a wide range of natural phenomena, and so a theory of them is widely useful.

But usability of a mathematical model leads to enthusiasm and enthusiasm may lead to completely ignoring niceties such as the relationship between the model and reality.

We could go with the flow and allow the model definition of information to override our previous understanding, and declare that information is negative log likelihood of selecting a particular message from a finite set of messages. It's tractable so it's true. After all, no-one likes to be stuck with intractable problems. Sundman's result was big at the time, but no-one remembers him now.

A mathematical model is a structure. A particular structure is picked out because it is user-friendly and because an isomorphism exists which maps onto it something that is close to something we are interested in. Shannon information is tangentially close to real information: there's a vanishingly small place where it's an exact match, a fuzzier area where it's very close and lots of places, such as en route to Barnard's Castle, where it's just wrong.

It seems an elementary mistake to say, “this structure lets us calculate stuff and therefore reality is exactly like this” yet it happens all the time. As people learn a result calculated using information theory, or Turing computability, or Bayes theorem, they go into the world convinced they're seen the light.

They've seen the light of a structure that is isomorphic to something that is close to something that interests us. They may never notice the assumptions and definitions required for the structure to count as a map of reality.

The isomorphism is not the terrain.

A Supernaturalist and a Physicalist Swap Accounts of the Universe

A supernaturalist S and a physicalist P swap accounts of the existence of the world.

The supernatural account (S) of “something exists and behaves reliably enough for us to theorise about it” might go like this:
S: Something exists that is eternal and able to create a reliable universe which we can theorise about.
P: “How can that be?”
S: I don't know, it's supernatural.
P: Well that's not very satisfactory! Here's my account.

The physicalist account (P) of “something exists and behaves reliably enough for us to theorise about it” might go like this:
P: Something exists that is …[insert P's detail here]…
S: “How can that be?”
P: As a physicalist, I have a proper explanation. Look, here's my explanation, E, showing how it follows mathematically from certain equations and conservation laws.
S: “Nice. 2 questions though. (1) May I read your explanation E? and (2) does it include an explanation of how come these conservation laws and equations hold?”
P: Regarding (1) no you can't see explanation E because no-one has written it yet. And (2) it probably won't include that, we usually just accept them as a given.
S: Ok. Let's pass over the fact that no-one has written explanation E yet. When someone does write E, the account of “these conservation laws and equations hold” will be supernatural. Look, here's my account:

S2: Something exists that is eternal and able to create a reliable universe which instantiates these conservation laws you and conforms to these equations you mentioned.
P: “How can that be?”
S2: I don't know, it's supernatural.
P: Well that's not very satisfactory! Here's my account.

P2: Something exists and it instantiates these conservation laws and conforms to these equations we mentioned.
S: “How can that be?”
P2: It just does. It's a brute fact. That's the theory.
S: What makes this account physicalist rather than supernatural?
P2: Because I don't invoke anything supernatural.
S: What's a brute fact if not something supernatural? You explicitly use it to mean those phenomena for which you offer no physical account. Otherwise we wouldn't be calling it brute fact. So it's super-physical. Or, as we say in everyday English, supernatural.
P2: That is not what we usually mean by supernatural.
S: Hmmm. I thought “not explained by physical laws” was what we meant by supernatural?
P2: You can't expect the physical laws themselves to be explained in terms of physical laws.
S: You're right, I don't expect that. What I expect is, to recognise that saying physical laws can't be explained in terms of physical laws means they are inexplicable (in physicalist terms at least), and hence anything that a moment ago you explained in terms of them is, also, inexplicable in physicalist terms. There is no physicalist account of the world we live in, only supernatural ones.

Pretending that SQLite has Stored Procedures and Functions

SQLite is marvellous. The fact that it doesn't have SQL syntax for stored procs and functions is usually not a handicap because it has an interface for the consuming application to register functions, which means you get to write your functions in your preferred programming language. Win all round.

But sometimes you do wish you could do it in SQL.

The lack of Stored Procedures is usually ok—you can just use scripts. Variables are easy to do: create a one-row temporary table and call it args or var. Working around the lack of Functions seems harder but in fact, you can program functions with Views. You can use CTEs in a View definition, so you can build up complex calculations. And CTEs allow recursion so you have Turing completeness.

As an example, the Exponent function as a View:

Drop Table if Exists Args ; Create Table Args as Select 5.5 as Base, 4 as Exponent ;

Drop View If Exists Power;
Create View Power As
    WITH RECURSIVE pow(exponent, exponent_remainder, base, result) as (
        --FIRST EXPRESSION
        SELECT exponent, exponent-1 , base, base
        FROM Args

        union all
        --SECOND EXPRESSION
        select Args.exponent, pow.exponent_remainder -1, pow.base, pow.result * pow.base
        from Args
        join pow on Args.exponent = pow.exponent
        where pow.exponent_remainder >= 0
    )
    select pow.result
    from pow
    where pow.exponent_remainder = 0;

and now you ‘call the function’ with:

Update Args set Base=2.5, Exponent=5; Select Result from Power;

The elements of the workaround are:

  1. A one-row table for function arguments
  2. A view which can refer to the arguments table and do the calculation. Since you can use CTEs to do recursion, you could in principle programming anything this way.

In similar style, here's an Exponential function which lets you specify how many significant digits you want the result to, default to about 7 digits of accuracy. This time we call the Args (X,Y,Z,p4,p5,…)

Drop Table if Exists Args ;
Create Table Args as Select 1 as X, 2 as Y, 3 as Z, 4 as p4, 5 as p5, 6 as  p6;

Drop View If Exists Exp;
Create View Exp As
    WITH RECURSIVE exp1(X, N, term, approx, accuracy ) as (
        --FIRST EXPRESSION
        SELECT X, 1, X, 1+X, Max(Min(Y, 1),0.00000000000000001)   FROM Args

        Union All
        --SECOND EXPRESSION
        Select X, N + 1, term * X / (N + 1), approx + term * X / (N + 1), accuracy
        From exp1
        Where  term / approx > accuracy Or N <3
    )
    Select approx as Result From exp1 Order By N Desc Limit 1;

And then:

Update Args Set X=22.0, Y=0.00000000000001;
Select * from Exp;
#
# > 3584912846.1315813 # Exp(22) correct to 14 digits

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