I’ve made mistakes in life. Many have been the result of too little coffee, too much alcohol or not wearing clean underpants. But some have been mistakes of logic. I want to stop making the logical mistakes at least, so I’ve been thinking. Here’s where I got up to:

Every way we have of reasoning is an abstraction. In that process of abstracting, something is lost. We can reason our way to things that aren’t true in real life if what was lost was important…

**1. Actual models**

I like to think about models of reality like matchbox cars, architectural drawings, rubber-band aeroplanes.

These things are simple abstractions. They abstract in useful ways, and their limitations are really obvious.

A matchbox car is not a car. It’s a simple version of one, designed to usefully demonstrate two of the car’s features: its ability to roll, and a scale version of how it looks. Noone goes on to think cars have plastic wheels.

Just as an architectural drawing is not a house but a useful model of one. It has visuals from several angles, materials, measurements. Its usefulness is for building a house.

A rubber band aeroplane has thrust, has lift, can take off. It is good for demonstrating the physics of flight.

We can do things with the model that we can’t do with the real world, like make a plan for a house that’s ONE MILLION STORIES TALL!!

Or we can drop a whole shoe box full of matchbox cars from ten stories and not have them crushed. No-one would object to these models describing a world that is not true, because the models are understood to not perfectly reflect the real world.

**2. Economic Models.**

For example, the supply-demand model. It abstracts enormously, from millions of disparate human interactions which happen in richly complex personal and cultural surroundings, to two dumb lines on a graph.

We acknowledge that the details we omitted are relevant, but not so relevant they void the model’s conclusions entirely. The model is imperfect and wrong, but useful to the economist in his ivory tower, because of its ease of use.

A supply and demand model is no good for finding out why people chose to part with their money, whether they caught the bus to place where they bought it, or whether they experienced buyers remorse when they got home. All it is good for predicting is prices.

We are aware of the model’s limited usefulness. We are aware of the assumptions underpinning the model – freely fluctuating prices, rational consumers, competition, diminshing marginal costs. If those don’t hold, we know we shouldn’t trust the model.

We know we can do things with the model that we couldn’t do with the real world. We know we could show prices less than zero, upward sloping demand curves, etc, but because such dumbness would be transparent, we avoid doing so.

**3. Descriptive Language**

You can describe a building in a way that, like a model, cuts out superfluous details. E.g. Tall, sandstone, located behind the library.

Language is so adaptable and fluid, that it is hard to remember that it is like a matchbox car version of the real world. It doesn’t catch all the details. A lot of information about the building is not expressed.

But even those aspects captured might not be as unambiguously expressed as a matchbox car expresses the shape of a car. All language depends on a link between name and object being understood. *Library* and *sandstone* may be obvious. But many things are not so clear. *Tall* is relative, *behind* is relative.

The description of a tall sandstone building is more like a child’s drawing than an architectural blueprint.

**4. Arguments**

If descriptions are bedevilled by omitted fact and subjective interpretation, then this shortfall of language becomes damn pernicious when we describe logical processes, i.e. arguments.

One problem is that unlike a matchbox car or an economic model, there is no general agreement on the way in which language omits potentially relevant details. The real world is so rich in detail that we are obliged to take it as given that the 99.999% of details the writer or speaker has omitted were not the relevant details.

*“If A, then B*.” The logical linkers – if and then – work on the words not the real-world concepts behind them. If if and then are misused, or *A* and *B* are ill-defined, we can reach false conclusions.

Here’s a sentence from Michelle Grattan’s article today in The Age.

“We’ve come to this predicament because Labor performed poorly, but not poorly enough to be dispatched directly out the back door, while Abbott did well, but not well enough to get there in his own right.”

This is a logical statement of the form “A because of B.” **A** is “this predicament” **B** is “Labor performed poorly; Abbott did well.” Both these reasons are qualified.

The qualifications show that the term “poorly” is not really defined, or irrelevant, and the same goes for the term “well”.

The “because” in the middle of the sentence is a bold step by Grattan. Why are two causes given? Would each be powerful enough on their own? Was each cause necessary but insufficient? We are not told. Are all other reasons considered irrelevant? What is the role of the independents, the media, and the Greens in “this predicament”?

Labelling also causes problems here. Why is it Labor and Abbott not Labor and Liberal? Are Hockey, Robb, Truss and Bishop excluded from Grattan’s description of causality?

It seems that while this beautiful sentence by a master analyst maps onto a vision in our brains, in the end it tells us nothing certain about what happened, nor anything about any causes or effects.

No offence to the superb Ms Grattan is intended. I hope this one simple example has helped show making sense will depend on having a shared understanding of the ways in which language obscures its assumptions and omissions.

And yes, I’m aware of the irony of arguing in words that arguments in words are generally wrong.

**5. Numbers.**

So far, anyone that’s waded down this tortuous, piranha-infested logical river with me hasn’t found anything they didn’t already know or think of themself.

So here’s my final point.

Numbers suffer from the same problems.

1. Numbers depend on abstraction. We can only number what we can name.

2. The link between the number system and reality is only partial. Zero and negative numbers exist only in the number system. So do imaginary numbers.

The first point may seem trivial, but consider these examples:

There are fifty matches in a box. But if you light one and put it back in, how many do you have now? Most people would now say 49, on the basis that the defining part of the match is the sulphurous head. But, previously if you’d showed them a burnt match they would likely still have called it a match.

Fingers might seem discrete until you accidentally lop one off at the second knuckle. Then you might say. “I cut off my finger. Now I only have half a finger.” Wait, hang on, was it one or a half?

We rely on the number system for many many things. It works very well for things that we can number, especially things that are themselves abstractions, like dollars, hours and years. It even works well enough on things like matches and fingers if we expend adequate language on defining what we are talking about.

But there are areas where we rely on the number system that are outside the conceptual field in which our naming and counting conventions developed. Therefore it may be that numbers will not work on sub atomic scale, where all is indsintiguished quarks, and universal scales, where all is a single entity.

I suspect that this might explain string theory. You can use maths to prove anything, but if you are applying the number system to something you can’t define, you may be applying a system of numbers to something it doesn’t work on. The answers found may come to more reflect the system used than the physical reality being studied.

It would all be a lot clearer if you were Vulcan…

You lost me at the million story tall building. Where would it go? Would it have balconies? I ask, because at a million stories, there wouldn’t be much air. In fact, you would need pressurised apartments. Or offices if it was zoned commercial.

would you believe i’ve done the (spurious, subjective, unreliable) maths? A million stories is about 3,500,000 metres, or about a tenth of the way to the moon.

You are 3,400 km above the generally accepted start of space, and in the fastest elevator in the world (only 66kph, turns out), it’d take 2 days, 3 hours and a couple of toilet breaks to reach the top storey, assuming no punk kid had pushed every button in the lift, in which case you’d probably die before you got to your 10 o’clock (next Friday) meeting…

Which brings us to the space elevator. Not entirely infeasible it would seem, except for this little excerpt from wikipedia:

Impacts by space objects such as meteoroids, micrometeorites and orbiting man-made debris, pose a more difficult problem, because the potential of a strand break to cause a failure cascade is, according to Tom Nugent, the Research Director of LiftPort Inc., “A potential show-stopper for construction of the space elevator [that] has not yet been adequately addressed.”

well, thanks a lot for engaging with the content of the post, everybody.

That’s because I feel that this is a topic best discussed over a serious number of bottles of red wine, using as much emotive, imprecise, incomplete argument as is possible.

Written words (such as a reply to a blog post), along with the aforementioned problems up there in 3 and 4, have the added problem of being frustratingly unchangeable, whereas the flexibility of verbal argument (as anyone who has had the distinct displeasure of engaging in semantic disagreement with young Dufus will attest) has the advantage of allowing each side charging headlong towards heated, passionate, violent, grudging agreement over the course of several agonising hours of sparring.

I wonder how much one of the apartments on floors 901,000 to 999,999 would cost…

I think your meditation is spot on T’E, but I was surprised to hear that people accost you for saying numbers are subjective. As I understand it, you refer the to ‘gap’ between what we observe and what is, in reality, extant as being the point where numbers fail us.

I think using the number one is a good starting point. Mathematically it’s pure, indivisible, perfectly defined truth. But even in our conceptual field (as opposed to the fields beyond our conceptualisation to which you refer) it is flawed in application. Simply because the way we separate ‘one’ from ‘another’ is always arbitrary even in simple scenarios. Where does the finger end and the hand begin when you hold a fist?

But before we dive head first into an infinite regression we must acknowledge how powerful and effective the concept of 1 is, hello digital watches. In these situations I like to refer to one of my favourite philosopher’s, John Dewey (yes, of Dewey decimal system fame), the pragmatist who, allow me to severely paraphrase , stated that truth can be found in what is useful. Which at least works for me when I’m counting matches.

Thanks for engaging Johnny B.

In response I can tell you that the man who valued the useful was not the man behind the useful system.

http://en.wikipedia.org/wiki/John_Dewey born in Vermont 1859

The great cataloguer was melvil dewey http://en.wikipedia.org/wiki/Melvil_Dewey born New York 1851.

I only know because it is a mistake i have made myself.

Really! Shit.

OMG. String theory is not the sort of thing you can shoot the breeze on, with no expertise whatsoever.

“2. The link between the number system and reality is only partial. Zero and negative numbers exist only in the number system. So do imaginary numbers, but we use them to understand reality.”

What is “the number system”? The term “number system”, let alone “the number system” is somewhat vague in mathematics.

A number system is more or less understood to be a system that shares ring-theoretic or field-theoretic properties with the set of all real numbers, equipped with addition and multiplication. Z, R, C, equipped with the appropriate operations (addition, multiplication, etc.) are different number systems. C contains R contains Z.

There is no qualitative difference in the legitimacy of i (square root of -1) and 1, 2,… etc. The complex number i is conceptually less concrete, more abstract, compared to the real numbers, that’s why it’s called “imaginary”. But its actual existence is no more or less dubious than one, two, sixty-three, the Frucht graph, the alternating groups, or any other mathematical object.

The layman such as yourself probably cannot see that questions like “what is i?” or “what is a complex number?” are meaningless.

Have you ever heard of coordinates? Cartesian (two-dimensional) coordinates are ordered pairs of real numbers, things like (0, 0), (1, 0), (pi, -1), (1/e, e^2), etc. [Here e=2.718 approximately is the well-known irrational constant].

Complex numbers (and imaginary numbers) are nothing other than Cartesian coordinates. That is the simplest definition. Primary schoolkids know what these are. It’s the multiplication that one *defines* on the complex numbers that gives them their apparently mathematically remarkable properties.

Let (a, b) and (c, d) be Cartesian coordinates. Define (a, b) + (c, d)=(a+c, b+d), and define (a, b) x (c, d) = (ac-bd, ad+bc). The set of Cartesian coordinates together with the operations “+” and “x” just defined constitutes the complex number system. Or rather, that’s one way (the simplest) to define the complex number system. And in fact that’s the sensible question to ask: “What is the complex number system?”

In mathematics it’s often more important to know how things are related to each other, than to know what things actually are. In fact this latter question “what is it?” doesn’t always make sense. Unlike in the natural sciences or engineering, or most real-life situations.

So the set C of all complex numbers does not do much as a number system. Rather like the set N={0, 1, 2, …} of natural numbers does not do much as a number system, without also appending the operations +, x, and perhaps the induction axiom depending upon the context.

I hope that clears up somewhat the nature of the various number systems used in mathematics, including the complex numbers. If it raises more questions than it answers, well, that’s still a good thing. Because then, at least you know that you don’t know. Whereas before, you didn’t know that you didn’t know.

“But there are areas where we rely on the number system that are outside the conceptual field in which our naming and counting conventions developed. Therefore it may be that numbers will not work on sub atomic scale, where all is indsintiguished quarks, and universal scales, where all is a single entity.”

N and Q+, the natural numbers {1,2,…} (or {0,1,2,…}) and the set of all nonnegative fractions, are fairly intuitive and concrete, and perhaps in some sense “came” from everyday caveman needs. Nonnegative irrational numbers also came from geometric problems and lines in the sand. Other number systems were developed due to more algebraic and abstract needs.

The fact that mathematical tools that were developed to solve everyday problems evolved (in the arena of mathematics) to the level where they were fitting to subatomic physical problems, as well as the global structure of the universe, is remarkable. It’s remarkable because your argument, though plausible, has a conclusion that has been disproven time and time again (relativity, quantum mechanics, thermodynamics). One could look closer and question why this is, or whether it will remain true. I believe it’s called “looking a gift horse in the mouth”.

“I suspect that this might explain string theory.”

It doesn’t explain string theory.