May 1, 2011

Logical Paradoxes

I have not written anything for a while, and part of the reason is that I have not received much feedback on what I have written so far; if you like this blog and want me to write more blog entries, please let me know.

In the hope that it might interest people, I decided to write a blog post on logical paradoxes such as the following one:
Sentence S: Sentence S is not true.
Suppose S is true, then it is not true. Suppose instead it is not true, then it must be true...

Or take another example:
Sentence T: If sentence T is true, then 1+1=3.
Suppose T were true, then (think carefully about it) 1+1=3. But from what I just said it follows that T is true (why?) and so it must be that 1+1=3 (why?).

By reflecting upon sentences such as S and T one can (as you hopefully just saw) come to absurd conclusions. In practice, everyone sees that something must have gone wrong and forgets about the whole piece of reasoning, but one might want an explanation of what exactly it is that goes wrong.

It seems to me that the paradoxical aspects of S and T can also be found in paradoxical descriptions such as the following one:
Description D: The natural number 1 – δ(D).
To understand this description you first have to know what I mean by δ:
Definition of δ:
 δ(x) = 1 in case description x describes the natural number 1, and
 δ(x) = 0 in all other cases.

(Why is D paradoxical? Suppose D describes the natural number 1. Then δ(D) = 1. So D simplifies to "The natural number 1 – 1". Which simplifies to "The natural number 0". But then D describes 0, contrary to our assumption. Suppose instead that D does not describe the natural number 1 and you will get another contradiction in much the same way.)
With sentence S one might think that the problem lies with the word "not" or the word "true", and with sentence T one might think the problem is related to the construction "if ... then ...", but one can hardly argue anything similar with D. Yet, it seems to me that the paradoxical aspects of S are also found in D, so what exactly is wrong with S, T, and D?

One question we have not considered so far is whether we are at all able to make sense of S, T and D. Like the mathematical expression "0 / 0" they combine familiar things in an unusual way, and since 0 / 0 is often said to be "meaningless", perhaps we should not be so quick to think that we can give a meaning to S, T, or D. I personally think 0 / 0 can be given a meaning (see but if you ask me whether S (or T) is true or what natural number description D describes, then I have to say I do not know how to make sense of the question (in the above I was reasoning in a naive way for demonstrative purposes). If you think you can make sense of these questions, would you please tell me what their answers are? And if you say you do not know the answers, could you at least give some indication on how one might go about and find them?

April 2, 2011

Mathematics, Logic, and Computation

This time I intend to explain how I think about "mathematics" and "logic".

I have no problem speaking of "mathematical truths", like "the truth that every natural number has a unique factorisation into prime numbers". I also have nothing against speaking of "mathematical objects", like "an n-dimensional space of constant curvature k". Furthermore, I am happy to speak of "ideas" of mathematical objects, like "the idea of a triangle". In fact, I think Plato deserves a lot of credit for introducing "the world of ideas/forms".

Yet, I also embrace some aspects of what might be considered the opposite of Platonism: Constructivism. While Plato thought of mathematical objects as "eternal", constructivists say that mathematical objects are "constructed", where that construction is something that happens in time.

So how can mathematical objects be timeless and yet be constructed in time? Think of fairy tales: Writing a fairy tale is something that takes place in time, but the fairy tale itself need have nothing to do with our time. In a similar way, one can construct a mathematical object that has nothing to do with our time.

But there are further twists to this. The fairy tale does have its own, "internal" time (inside the fairy tale, one thing happens first, then another happens, and so on), and in the same way a mathematical object may have its own, internal causal structure. To take a somewhat extreme example, the object could be a "space-time".

For another twist, imagine a fairy tale that includes a person who writes fairy tales and constructs mathematical objects. In a way, the "construction" of mathematical objects can happen "outside time".

Although I have now spoken of people and fairy tales, I may as well speak of computations. For example, just as one fairy tale can be written inside another fairy tale, so one computation may be realised within another computation. Mathematical objects may be timeless, but they certainly arise in computations, and you can make beautiful pieces of mathematics come out of computations (look at some fractals if you are unconvinced).

But what are those timeless "mathematical objects" and "mathematical truths"? Read my blog post Objects as Information to get part of the answer. I think of mathematical objects (and mathematical truths) as "pieces of information" that can be stored in computer memories and used in computations.

But what exactly do the pieces of information in question look like? I will return to this in future blog posts. Suffice it to say for now that if you catch a mathematician talking about a mathematical object you can ask the mathematician for "axioms" describing the mathematical object. Those axioms give you a good starting point when you want a piece of information that completely captures the mathematical object. As you interview the mathematician you may, however, find that the mathematician lacks a clear idea of what exactly the mathematical object is, and so you should also look at how it is used and what intentions the mathematician has when using it (I am reminded of my blog post Meaning and Goal-Directedness). Looking at how objects are used is also a key to identifying mathematical objects inside computations: Look at the functional role played by information at various points in the computation.

Leaving to future blog posts the question of what exactly mathematical objects look like, let us now instead think about mathematical truths. What on earth does it mean that one mathematical truth "follows from" another mathematical truth? To take a simple example, by what right can one say that "from x=y and yz it follows that xz"? Well, "x=y" and "yz" are pieces of information (see my post Objects as Information), and the question is whether these two pieces of information when combined somehow contain the information that "xz". Now, "=" and "≠" can each be thought of as being (more or less identical to) a piece of information, and these are contained in "x=y" and "yz" respectively (what this means in practise is that you can ask a person who tells you that "x=y" what he/she means by "=", and likewise for "yz"). From these pieces of information you will get the information that "from x=y and yz it follows that xz", at least unless someone is using the signs "=" and "≠" in a very non-standard way.

So the "follows from" of logic is really the "is contained in" of pieces of information. Obtaining new pieces of information from old ones is a hallmark not only of "logic" but also of "computation", and I would say "logic" is really about computation/causality.

In a simple computation things proceed one step at a time ─ having performed step n you proceed to use the result of that step to perform step n+1 ─ but in general things can be much more complicated, not just with computers but also in mathematics and in physics. It may make little sense to speak of the cause for something. Rather, truths may be "over-determined", "true for more than one reason".

Well, this was at least an introduction to how I think about "mathematics" and "logic". In future posts I will discuss "symmetry", "definability", "identity", "truth values", "relations", "parts", "ideas of objects", "variable objects", and other stuff which, while central to mathematics and logic, would have made this post way too long.

March 20, 2011

Introducting Things, Making Identifications, Referring to Things

In ordinary speech we can "introduce" things by using phrases like the following:
  1. Once upon a time there was a ... who ... .
  2. Imagine a ... such that ... .
  3. Let P, Q, and R be the vertices of a triangle in which all sides have the same length.
  4. Consider a country which ... .
In addition, we can make "identifications" among things that have been introduced using phrases like the following:
  1. Let us say that person A and person B are one an the same person.
  2. Let us say the person we are talking about is me and that room we are talking about is the room we are in right now.
  3. Let us say that the radius of circle C is actually 3m.
  4. When I say "forward" I mean the direction in which I am now pointing.
We can also do what may be considered the opposite of introducing things: We can "forget about" things. We can do so by saying things like: "Let us forget about X and concentrate on Y" (in addition, we often "forget about" things without being explicitly told to do so).

Similarly, we can do what may be considered the opposite of identifying things: We can "make a distinction". For example, we may say: "Let us distinguish between 'The person who performs role A' and 'The person who performs role B'. In what we have said so far we have assumed these to be the same, but let us imagine situations where they are not."

Actually, the things just listed need not happen through language at all. Nature itself may "introduce new things", "make two things become one", "make an object disappear", or "split an object into two". And in a fairy tale, things can be "introduced" and made to "disappear" though magic. With computers one also "creates" ("introduces") and "deletes" ("forgets about") objects: For example, I am right now creating a blog post.

What I am getting at is: The operations I have described are very fundamental. Arguably, you find them in any computation, wherever you have cause and effect. They are certainly central to the contents of this blog. Whenever you see questions about what words mean/refer to (as used by some particular speaker in some particular situation), think to yourself that this is really about introducing things (words/concepts as well as things they refer to) and making identifications.

One type of identification deserves special attention: We may identify something we have "introduced" with something with which we can interact. This is very relevant to the questions about "how language is connected to the world". To deal with problems of this kind, imagine "a human-like being B in an environment E". You can then discuss how B interacts causally with E and how words that B uses are linked through causal chains to things in the environment E, but then you can either:
  1. Identify E with your own environment (and, optionally, B with yourself). This makes what you said about B and E "become real". It takes you from "mere ideas" to "actual things". 
  2. Make assumptions on E that means it cannot be "our world" (for example, you can assume it has a number of dimensions that differs from what we are used to). The possibility of doing this means that noone can tell you that B and E must be in a certain way. You are free to see them as your own fictive inventions which you are free to shape in your own way. Make appropriate choices, though, and it will be undeniable that what you say is relevant to "reality".
Having written this, I realise that "connecting language to the world" is just like connecting one machine to another, or like connecting two computer programs. Wittgenstein made this point, and I wrote about it in Meaning and Goal-Directedness. Connecting machines and computer programs is actually a perfect example of identifying things, and "connecting language to the world" is like "connecting a machine to the world".

March 16, 2011

Objects as Information

The blog entries of this blog are perfect examples of objects that may be thought of as "pieces of information". More generally, anything that can be stored on or transferred between computers may be thought of as pieces of information.

Not every object is of this kind, though. For example, the window in front of me is nothing I can upload and make available for download on the Internet. I can, however, take photos of the window and store those photos in a computer, and I can describe what the window looks and store that description on a computer. In fact, I could make my description so detailed that you could construct a close-to-identical window, and so it certainly seems that while I cannot store the window on a computer it somehow nevertheless involves information.

An example that falls somewhere in between the two types just exemplified is money. Today money is stored on and transferred between computers, but there are restrictions on how money can be "copied". Money certainly involves information, though.

Must an object inevitably involve information? I think the answer has to be yes, because we can always ask questions about an object, and the answers to such questions give us information about the object.

There are several points that can be made on the basis of these considerations. First of all, with two pieces of information we can ask whether one "is part of"/"can be extracted from" the other, and if objects are (closely related to) pieces of information then this carries over to objects. For example, the first paragraph of this blog post is something that can be extracted from this blog post.

Second, if objects are closely related to pieces of information, then what about truths? It is a truth about this blog post that it contains more than four paragraphs, and this truth can surely be seen as a piece of information. In fact, it is a truth that can be extracted from this blog post (given the terminology just introduced).

Third, is not a (possibly incorrect) statement about an object also an example of a piece of information that is related to the object? With a suitable meaning of "statement" the answer must be yes. For example, if I say that "The window in front of me has a blue frame", then that is a piece of information I extracted from an object, but it can also be looked upon as a possibly incorrect piece of information (statement). To see if it is correct or not, examine the object it purports to be about, see what can be extracted from that object.

March 10, 2011

Certainty as a Security Issue

One of the classical problems of philosophy is how we can be certain about things. Could it not be that I am right now dreaming, that the things I see around me do not really exists, that my memories have just been fabricated by someone who wants to delude me?

When dealing with problems with this one, the first thing I do is do generalise to arbitrary human-like beings: How can a human-like being at any one point know that he/she is not dreaming, etc. ? Generalising in this way is not necessary, but I believe it is a good practise; it may help us avoid obsession with ourselves.

I never said that the human-like being had to be real rather than fictive, so take your favorite fictive character and imagine him/her asking whether some object he/she is interacting with really exists. Assuming that fictive beings do not exist, the answer may be that the object in question does not exist. But of course, all this means is that the question should be reformulated: How can a human-like being know that some object it is interacting with is not just an illusion?

Let us now admit something that should be trivial and uncontroversial: Human-like beings can be dreaming, they can discover they thought they were interacting with some object when actually they were not, and they can misremember things. So let us rephrase the question: How is it that under ideal circumstances a human-like being is able to be absolutely certain it is not dreaming or being deluded in some other way?

I have come to believe that one should approach this question as a security problem. Just as we can guard ourselves against all sorts of crimes, so we can guard ourselves against individual or collective delusions. Or, to put it in general terms, someone/something that has/may have knowledge can take security measures to defend itself against delusions.

We are living in a world full of risks that we can never quite eliminate, so let us approach the problem of absolute certainty with some humility. To obtain a truly well-founded security we ought to think the way security experts do, but instead we tend to incorrectly feel secure in situations where a security expert would see risks, and at the same time we tend to worry about things a security expert would not worry about.

That said, just as we can often be reasonably confident that an object we have stored somewhere is not going to be robbed within the next hour, so we can often be reasonably confident that we are not going to fall prey to some big delusion within the next hour.

By the way, security thinking and knowledge are things that apply to organisations: As an organisation interacts with its environment it should be guarded against all sorts of threats, including things that can make it think it knows things which in reality it does not know.

March 9, 2011

Meaning and Goal-Directedness

Welcome to this blog. This is my very first blog post, and writing it feels a little strange since I do not yet have an audience, but hopefully I will get one...

I recently re-read Ludwig Wittgenstein's Tractatus Logico-Philosophicus for the n:th time (I have got no idea how many times I have read it; certainly many more times than any other book) and then decided to go on and read something from Wittgenstein's later writings (with which I have considerably less aquaintance). Only having The Blue and the Brown Books on my bookshelf, I started reading the Blue book, and as I did so I started thinking on my own about the basic question: What is the meaning of a word?

Wittgenstein notes the importance of how words are used: "But if we had to name anything which is the life of the sign, we should have to say that is was its use." Using a word is a special case of doing something, of participating in an activity, and this should make you think of "causality" and "information transfer". How exactly do signs/words fit in here?

One may start by observing that a sign may be used "as opposed to" another sign. For example, a spot on a paper may be black "as opposed to" white. This is obviously related to information/alternatives/possibilities (knowing that the spot is black rather than white gives you one bit of information). Moreover, as Kripke argued in Naming and Necessity, names are connected to the things they name through causal chains. Finally, we may observe with Wittgenstein how words/signs/names perform functions (another word that should make you think of causality) in particular situations (or "language games", as Wittgenstein used to say).

The word "function" makes me think of technology, and unlike Wittgenstein I do not normally think of cogwheels and other mechanical things, but about computers and connections between them. It seems to me that computers offer excellent examples of "language games". For example: Imagine a computer program with one button saying "Gray" and another button saying "Black". Pressing either button will cause a printer to print out a spot that is gray or black according to which button was pushed. The printer is connected to computer B, but to press the button you need to use a mouse or a keyboard connected to computer A (physical connections through cables and the like enable "causal connections" -- once again, causality is what is important).

I guess it would make sense (even though it would certainly seem odd at first) in this situation to say that "the meaning of the button labelled 'gray' is a gray spot on a paper", and one could similarly say that "button 'Purple' and button 'Violet' have the meaning" or that "button 'X' has not been assigned any meaning". In any case, meaning certainly has to do with causal connections.

However, if you are feeling that we humans are not like machines when we "mean" something, then I think you are right, and it seems to me that Wittgenstein misses something important here: Humans may be seen as "organisations" of sorts, and the meaning of a statement made by an organisation is often related to the interests of the organisation. For example, when one "says A but means B", the way to know that one means B is to look at what one is trying to do (for example, one may be trying to win a game, or one may be struggling to avoid being one of the losers in the game). Note that someone who knows what one is trying to do may automatically understand one as meaning B rather than A even though one actually said A.

For theoretical work on "organisations" one may look at subjects like "organisational theory", "operations management", "systems theory" or "cybernetics". I am mostly a novice here, but organisations are so important that I will need to write about them again and again anyway (for example, knowledge can be thought of as always involving an "organisation" that has the knowledge). If you think there are things I am missing or stuff that I ought to read, just write me an email or leave a blog comment.