The seed of reason

What constitutes valid criticism of a scientific model?

The subject of this blog post is "What constitutes valid criticism of a scientific model?" (in relation to its predictions).  Certainly there seems to be much dubious criticism of scientific models particularly, but not only, in the mainstream media.  I'll give some examples in the form of thought experiments first and more concrete examples later.

By model I mean a simulation which is run in an attempt to derive predictions from a theory.  The first thing to note about such models, is that they are almost always inexact!  That is, the model is not completely accurate in its simulation of the scientific theory in question.  The reader might find it illuminating to search for counterexamples.

Models, however, can be more or less accurate, and certain models are closer to the underlying scientific theory.  Generally, models that are closer to the underlying scientific theory are better.  But as a practical matter we cannot demand that all models are completely exact because this currently requires (and probably always will) more computing power than we have available.

So the inexactness of a model on its own is not a practical way to criticize that model.  I give a few principles that I think any valid criticism of a model should obey.  Firstly though, I should point out that a model may be accurate in one area and woefully inaccurate elsewhere.  So we shall not be talking about criticizing a model itself but criticizing the model's use in making certain predictions.

Suppose that Alice has a model of some scientific theory.  Suppose in addition that Ben has found one or more inaccuracies in the way Alice's model simulates this scientific theory and Ben is claiming that this undermines certain predictions made by Alice from the theory with the help of the model.  Then my criterion (necessary but not sufficient) for Ben to have a valid criticism, are as follows:

1) The inaccuracy must affect the prediction in question.

Suppose Alice has a model of celestial mechanics that places the earth about one thousand kilometers further from the sun than it actually is.  Assume that the model is otherwise reasonably accurate in the size, shape, motion and spin of the earth, sun and moon.  This model will then predict the existence of seasonal variation in the apparent intensity of the sun.

If Ben criticizes Alice's model on the basis of its error then he is not making a valid criticism.

2) The inaccuracy must be significant enough to change the prediction in question.

Suppose Alice uses the above model to predict the timing of the tides.  Now suppose that Alice's model is used to calculate the tides for a given day, a month in the future, to a claimed accuracy of 1 hour.  The model will give a correct prediction, despite this error in the model.  The error is simply not significant enough, to make a difference in the calculation (the sun is many millions of kilometer from the earth).

Now suppose in addition, that Ben's criticism is intended to show that an alternative prediction is in fact correct (or more weakly that the correct prediction lies in a certain direction from that originally calculated).  Then I give further criterion for Ben's criticism to be valid:

3) A modified model must be given that corrects the given inaccuracy and this model must make a prediction which supports the alternative prediction.

Let Alice have a model of photosynthesis which overestimates the weight of a molecule of carbon dioxide.  Suppose Alice's model predicts the efficiency with which some type of plant converts solar energy into sugar.  Suppose Ben criticizes Alice's model saying that it overestimates the actual predicted efficiency.  It is reasonable to ask Ben for an alternative simulation showing that the known inaccuracy actually affects the prediction in the manner asserted.

4) The relevant prediction of the modified model must itself be robust to criticism on the basis of any inaccuracies the modified model still has.

Consider again, Alice's model of photosynthesis.  Suppose that this model, was also inaccurate, in its estimate of the weight of a water molecule.  Then suppose Ben, gives an alternative model, that is more accurate in its assumption, regarding the weight of a carbon dioxide molecule.  I can get a third model by adjusting Ben's model so that the weight of a water molecule is more accurate.  Now suppose my model makes predictions that are closer to those made by Alice's model than those made by Ben's model.  Then it is not clear whether Alice's model or Ben's model is closer to calculating the correct predictions as far as the original theory is concerned.  There may of course be further inaccuracies present in my model.

The above four points can be summarized as "You should trust the predictions of a model (assuming you trust the underlying theory) if those predictions are robust under various improvements of that model".  However, in practice, it may often be easier to identify invalid criticism, by referring to the above four points, in the order in which I've given them.

Of course, this also cuts both ways.  When Alice proposes her model, then she should estimate the error in the models calculations, by considering various improvements to her model, and seeing how much these affect the result of calculations based on her model.

In weather forecasting, many simulations are performed in an attempt to estimate the uncertainty in any given prediction.  Similar techniques are used to estimate the uncertainty in our global climate models.

Now for some more concrete examples.

A) Global warming is unlikely because climate models are inaccurate in X,Y and Z.

Many assumption underlying climate models may be gross approximations but the global warming skeptic needs to show that it is likely that these inaccuracies actually affect the prediction that releasing CO2 will cause global warming.

I am informed, by those who know more about the climate than myself, that this is not the case.

B) Consider a simplified economic model, that assumes no collusion between companies.  Then the price of X, will not be reduced by a decreased cost in the production of X as this model would predict, because companies will collude to prevent this happening.

This depends on what X is! The argument needs to be supported with actual adjusted models, that include some details of the game theory involved in company interactions.  The accuracy of the models, is best tested empirically here (checking them directly from quantum mechanics is not an option).

Pseudo-anonymity

Our personal data is being harvested for many purposes these days.  Stored in vast databases with little or no encryption organizations from our supermarkets to the security services are mining this data for reasons from commercial gain to national security.  How we keep confidential and sensitive information private is much debated.  However, important facts concerning cryptography are being ignored in this discourse.  Today I want to describe what is possible as a result of the revolution in cryptography that occurred at the end of the 20th century.

Our intuitions tell us that ineffective solutions aside (such as identification by indexed numbers) it is impossible to have both the benefits of anonymity and those of transparency.  But this is false.  Cryptography can combine benefits of anonymity and benefits of transparency.  Pseudo-anonymity is possible and comes in many forms.  Without an understanding of these possibilities any discussion concerning privacy will be be missing out on a huge range of potential solutions.

In what follows I will be making a couple of technical assumptions that are not hugely controversial.  Firstly I shall assume that certain widely believed mathematical conjectures are true or at least not usefully false.  I shall also assume that we are not going to be able to build quantum computers any time soon.  Our entire banking transfer system is based on these assumptions so I am in good company in making them!

Various counter intuitive things are possible with modern strong cryptography.  A cryptographic signature consists of a private and a public key.  Anyone in possession of the private key can sign messages (but no-one else can).  Anyone in possession of the public key can check the signature and read the messages.  Creating a cryptographic signature is easy with the right software.  Cryptographic signatures can be used to create a virtual identity which is hard (or if desired impossible) to tie to the person who created it.  However, over time such virtual identities can acquire trust in much the same way that individuals have for millenia.

Networks may be set up which allow anonymous communications to be sent.  This allows not only the content of a message but the existence of a message to be hidden.  Such networks already exist (for example Tor) and are used in high tech music piracy peer to peer networks.

Protocols are possible in which a certain action (such as decrypting a document) are only possible if certain people agree to the operation.  As an example it is possible to so encrypt a document so that any 3 people out of 5 key holders can decrypt it but that no 2 people can decrypt it acting alone.

A canary is a characteristic piece of data which identifies the source of a document.  The ordinance survey include minor errors dotted over their maps.  This allows them to detect any may which has been copied from an ordinance survey map.  Canaries can be added to many types of data to help identify unauthorized copies (although their utility is restricted to situations where few people have access to data).

No one has absolute privacy.  There are many ways in which your privacy may be intentionally violated.  For instance private detectives can be employed to follow you, public records mined for information or bribery used to obtain sensitive information.  We shouldn't try to get any absolute guarantees of privacy because we know it to be impossible.  In practice maintaining privacy is a matter of raising the cost of violating privacy to the extent that it is not worth the effort for the eavesdropper.

What matters is the cost of access to private data, the people who can access it, how easy it is to trace them and how susceptible the data is to abuse.  The problem with storing masses of credit card details in centralized databases is not that the information needs to be private but that the cost of steeling each record is lower by a kind of mass-stealing economy.  If only 2 or 3 people have access to a confidential file and anonymous blackmail threats are made then there is already a ready made shortlist of suspects available.  If furthermore there are tell tail mistakes in the blackmailer's threat (because canaries have been used) then the perpetrator may be identifiable.  Finally ease with which data can be abused matters.  There are sometimes alternative methods to store information and some may be less prone to abuse than others.

So taking all this into account we should worry when:

1) Data is stored in central databases

The more data in one place the cheaper it is to illegally access that data (per record)

2) This Data is in a computer readable format

Working through masses of data by hand takes many more resources than if that process can be automated.  As an example a supermarkets credit card database is computer readable but google street view is not (in any useful way)

3) Data is in abusable format

Records of transactions are necessary but these records can be stored in a manner which doesn't expose people's bank accounts to fraud.

4) The data is sensitive

Data about what music you like is not as open to abuse as data identifying which whores you've been visiting.

5) Many people have access to the data

It stands to reason that the more people have access to records the easier it is to trick/bribe them and the more likely it is that there are bad apples.

6) Its hard to trace the source of a leak

Clearly the easier it is to identify abuse the easier it is to discourage it.

7) The value of the data is high

A database containing movie preferences is much less valuable than one containing details of police cautions.  The second one needs much better protection than the first.

I will now give some examples of what cryptography could be used for.  Firstly it is possible to have electronic voting systems which are private but for which everyone involved in the process can count the votes themselves.  Unfortunately no system that is currently in operation uses the necessary technology.  Hence I am not against electronic voting in principle but I oppose all systems currently used.

Secondly any interaction that can be thought of as a sort of game with hidden information (such as a game of poker or a financial transaction) can be implemented using cryptography is such a way that the information can be hidden (such as the face of the card) the and yet when it is revealed the information is still known to be correct (revealing your hand).

Thirdly identity cards are possible which allow you to prove that you are a member of some group (such as non-terrorists or over 18s) without identifying who you actually are.  It is possible to do this without making the system more prone to abuse by terrorists or underage drinkers! Please note that the UK governments proposals for ID unbelievably do not use such technology.

The benefits of modern cryptography then are (A) that pseudo anonymity is possible are can be used to prove facts such as your age, your criminal status without revealing any other information (B) that signature schemes allow proofs of transactions without increasing the risk of fraud (C) that cryptographic protocols are stricter instruments of public policy than laws in that they can (subject to our assumptions) be mathematical proven to prevent abuses.  One of the many failings of modern liberal democracies is a failure to put our understanding of cryptography to work to provide these benefits and a failure to recognize the need for cryptographic solutions to provide privacy for the public, data for the government and intelligence for the police.

There are problems with cryptography too though.  Cryptographic protocols take time to perform but as computers get faster this objection becomes weaker and weaker.  Cryptographic protocols are brittle and not easy to adapt to new usage patterns.  I think that it is better to live with this than to risk the massive privacy violations that will occur without it.

Godel's incompleteness theorems and progress

Godel's incompleteness theorems were proved towards the beginning of the 20th Century and to this day they are amongst the most abused mathematical theorems around.  Today's post will have three parts:  Firstly I shall state what the first incompleteness theorem says.  Secondly I shall give you an example of how people attempt to apply it when talking of progress.  Thirdly I shall explain the hidden assumptions behind their arguments and give my own viewpoint on what the implications of Godel's work are for a belief in progress.

Godel's first incompleteness theorem states (this is not the original formulation) that no computer program can, unaided, list all correct mathematical theorems about the integers.

The argument against unlimited progress then goes as follows:

'Knowledge concerning integers and mathematical objects underlies much of progress in the sciences and even to some degree the arts & humanities.  So if there were a limit to our progress in mathematics then that would imply a limit to our progress elsewhere.'

Thus far I should comment the argument is fairly sound.  As mathematics deals with various abstract structures and instances of those structures can appear in other areas of human understanding it seems at least plausible that a failure to progress in mathematics should lead to stagnation in other disciplines.  The argument then continues:

'Therefore whatever physical laws our universe follows, since a computer could simulate them, we must have that the whole universe could be simulated by a sufficiently powerful computer.'

This in my opinion is a false step but I will finish giving you the argument before saying why:

'Humanity is part of the universe and so can in principle be simulated by a computer too.  Hence Godel's incompleteness theorem applies to humanity.  Hence humanity will never attain certain mathematical knowledge.'

The final part of the argument is also problematic but for different reasons.  I shall give criticisms of the argument and give my opinion on them:

1) '...since a computer could simulate them...'

This betrays the exceedingly dubious assumption that the number of physical laws is finite.  Without this assumption what follows in the argument does not follow logically.  Physicists are trying to come up with a general unified theory (a goal which was pursued also in the 19th century but ultimately without success).  However, it is by no means certain that such a theory even exists.

In fact the above argument really needs to assume that there is a General unified theory behind physics.  Otherwise consistent universes can be described where the argument fails.  I have more to say on the probability of this but that will have to wait for another post.

2) 'Hence Godel's incompleteness theorem applies to humanity.'

Strictly speaking we must have some way of extracting computationally the mathematical 'knowledge' that humans possess.  In an absence of this statement (2) is not meaningful at all.

This second criticism is not too biting.  The argument can just allow a skeptic to pick whichever computational method of extracting 'knowledge' she desires and the argument will show (assuming a GUT exists) that humanity cannot have access to all arithmetic truth according to that way of extracting 'knowledge'.

Finally what implications do Godel's incompleteness arguments have for a belief in progress?

A) If progress can continue indefinitely (and without limit) there must be infinitely many physical laws which cannot be reduced to some cleverly crafted finite set (so no GUT).

B) The structure of these laws must somehow encode the truths of arithmetic more over in a way which allows them to be discovered by appropriate means from within the universe.

(A) is not too hard to swallow but many may find (B) to be profoundly counterintuitive.  I shall return to look again at (B) in another post.

This post is part of a series investigating the belief in progress.  All past articles in this series are filed under the tag 'progress'.

Dangerous ideas II

Further highlights from "What is your dangerous idea?"...

6) Brains cannot become minds without bodies - Alun Anderson

Alun Anderson thinks it would be dangerous not to acknowledge this.  He's not batting for the existence of a soul here though.  He is trying to persuade us that it is not just our brains that generate our minds, but that our hormonal system, internal sensory systems, muscular feedback and spinal chord have important parts to play too.

Anderson argues that an understanding of the mind which encompasses more than just the brain is necessary to explain the placebo effect, reported physical sensations involved in the emotion love and links between stress and ill health.

Personally I think Anderson is onto something here.  I have noticed in the past that various emotions do seem to manifest themselves as aches, pains or internal sensations within my body.  Also it is clear that hormones, the spinal chord and other internal dynamics are important to mood, desires and can be used in problem solving (ever moved your arms to trace out a shape whilst thinking of it?).

7) Drugs may change the pattern of human love - Helen Fisher

Helen Fisher specifically questions whether a class of anti-depressants (Serotonin-enhancing) may interfere with the biochemical pathways necessary for romantic love.  Fisher provides evidence both that these drugs have such an effect and for a biochemical mechanism behind such an effect.

She regards it as a dangerous idea because if true it means that we are systematically preventing a significant percentage of our population from experiencing an emotionally normal life.  As Fisher points out it is also quite possible that social change and/or political changes may occur in our societies because of such an effect.

Although confronting such a dangerous idea may make one uneasy its clearly better to know the truth on this one.

8) You have the right to choose the sex of your child - Diane Halpern

Diane Halpern provides good evidence that left to their own devices families (and individuals) the world over have an inherent bias towards having male children (often through infanticide unfortunately).  Imbalance in the number of men and women is a problem (particularly in rural China) and various methods have been used to try to mediate this effect without much success.

The economist ran an article on the social effects that an imbalance of men and women had in China.  Amongst the problems were increased aggression amongst young males, increased rapes, forced marriages and a shortage of surrogate mothers not to mention larger numbers of men never marrying/having children.  This is a problem we should keep our eyes on.

9) The world may be fundamentally inexplicable - Lawrence Krauss

What is currently happening in cosmology and fundamental physics looks very much like what was happening in Chemistry before the periodic table.  Often in science just before a revolution in thinking more and more complex systems become necessary to fit new data into an old model.  I see no particular reason to think that we've reached the edge of our capabilities.  Krauss is delivering a council of despair before we've really got started trying to explain some new and interesting data.

Incidentally this structure of scientific revolutions (a weak reading of Kuhn) is interestingly similar to punctuated equilibrium (the hypothesis that evolutionary change proceeds in a series of slowly changing plateaus punctuated by sudden bursts of ultra-fast change).  The similarity is such that I'm tempted to hypothesise that there is a shared mathematical structure to these to phenomenon.

10) Insight is becoming impossible at the frontiers of mathematics - Steven Strogatz

Why Steven thinks this is dangerous is unclear.  It is probably undesirable.  Perhaps Steven thinks that the idea that this is happening is dangerous (he does think it is happening).

I think that the idea that this is happening is dangerous but furthermore I do no think that this it is happening (generally speaking I only think ideas that are not true are dangerous).  This sort of idea stems from the logical limitative results of the 20th century.  But these results show merely that absolute knowledge is algorithmically unattainable and that every system of logic is incomplete.

What seems to have been lost in all this is the possibility that progress in mathematics is nerve the less highly probable.  Again and again problems in mathematics that were considered unassailable have fallen (often to extremely elegant proofs).

Several things contribute to this.  Amongst others these are:  Advances in the expression of mathematical questions (e.g. algebra & calculus), advances in definitions (e.g. the definition of the group, the definition of first order logic), advances in communication between mathematicians (e.g. the journal, latex).

Such advances must not be underestimated.  The solution of the quadratic is much simpler once you understand algebra.  Computer visualisation programs can allow human pattern finding much more to work with.  Often those who cannot see the way forward decide we have reached a blind alley.  I suspect Steven Strogatz has made this perennial mistake.

There will be more highlights from this book in subsequent blogs.  Look out for the next instalment soon (I may blog on another topic first though).

The nature of existence

In this blog post I shall be addressing the question "What exists?".  This is a thorny issue and before going further I must warn the reader that my thoughts on this matter are in a state of flux (hence I may soon change my mind on these matters).  Never the less I shall present the state of my thinking on this.

1) There is some way of representing universes as mathematical entities (specifically infinite sequences in finitary languages (for mathematicians I am assuming AC)).  My basis for this is simple:  I regard mathematics as the study of all possible universes.

If mathematics does not allow the representing of universes (or at least their modelling for the questions in hand) then I regard mathematics to be incorrect or deficient in some way.

2) Two universes should be regarded as the same if their representations can be calculated from each other (in an objectively computational way).  If universe A is represented 01010201... and universe B is represented 0011001100220011... then there is a simple way to calculate one from the other & vice versa and so universes A and B should be considered identical.  This is a decidedly strong principle and should not be accepted without some deep thought.  It has some interesting consequences in ethics and aesthetics (perhaps another post topic).

This would imply that all finite universes are essentially the same (a very strong claim!).  I don't have any issue with this personally but you would if you believed that infinite universes were impossible (and perhaps for other reasons too).  It does not imply that all infinite universes are the same though.

3) There are two notions of existence.  Firstly what entities exist somewhere out there.  Secondly what entities exist that could in principle effect you.

3a) Consider all possible universes.  The question should be: "Which of these possible universes exist?".  Now if some possible universes exist and some do not we can ask "Why do some exist and others not?".  Either there is some logical difference between the contents of these possible universe or there is not.  If there is no logical difference then the fact of the existence of those universes is entirely arbitrary (A bad position I'd hope you agree).  If there is a logical difference we are asked to accept that the presence of some feature of possible universes makes them exist (say the property of having cats alive at some point in their history).  Again any such internal feature conferring existence if highly dubious.  It seems evident to me that under these conditions that in fact all possible universes should be taken to exist or all possible universes must be taken not to exist.  From these two possibilities I'd plump for the former!

3b) The second notion of existence can be defined roughly as follows:  Everything that has effected you or might at some future time effect you must be taken to exist.  Everything else does not exist.

Now for the punch line:  The second notion of existence is generally the most useful.  After all the tiger thats walking down the street outside your front door in a parallel universe is of no immediate concern to you.  However, when engaged in discussions concerning the probability of our universe the first definition is key.  Hence my answer to the question of why our universe has the apparent fine tuning it has is "All possible universes exist so there is nothing to explain."

4) Seeing a universe from a perspective is like projecting that universe down in some way.  Imagine the universe as a wire model and your perspective as the shadow it casts on the table top.  Different ways of representing a universe are different universal perspectives.

A word of caution:  This does not imply moral relativism.  The moral decisions you make must still be made from your perspective (although not necessarily with just any moral theory).

I may put up another post soon arguing why I think this attitude to the concept of existence implies that qualia don't exist and that the strong AI thesis is probably correct.

Complexity theory and certain common arguments

Complexity is the study of how difficult calculations are.  Some problems take a small amount of time to solve and some take much longer (using a computer).  Complexity theorists study the differences in a mathematically rigorous way.

There are many arguments I commonly hear which I think of as dubious.  Amongst these are some where the crucial error comes from a lack of understanding of the relative hardness of performing different calculations.

I shall give examples of instances where people fail to take complexity theory into account.

* In science different kinds of phenomena are explained using different scientific theories (from Quantum mechanics through natural selection to social choice theories).  Often its claimed that all you need to understand other aspects of reality is the base theory.  Strictly it is the case that (assuming we have a fundamental theory that is computable) we can calculate everything that happens in the world from the state of the world as it is now (albeit in probabilistic terms).  However the complexity of those calculations make it effectively impossible to do in many possible worlds (possibly including ones based on quantum mechanics).  This means realistically we do need other levels of science to understand the world.

* Any mathematical function from the integers onto the integers that can be performed can be reversed.  It used to be considered to be impossible for two people to talk in public (without having met before) and communicate information privately.  Strictly speaking it is always possible, given enough time, to figure out what it is they were talking about.  There may exist (assuming a certain prominent mathematical conjecture) functions that are much easier to perform than to do reverse (the mathematical equivalent of scrambling an egg).  This would allow people to talk in public without eavesdroppers understanding them.  Large sections of our financial institutions depend on this mathematical conjecture being true (or at least not nastily false).  It took some particularly brilliant mathematicians to see this was the case when the common sense view made it seem impossible.

* In order to protect private data we used to keep sensitive information in locked filing cabinets, keep the filing cabinets within corporate offices and employ only people who've been carefully vetted.  Some (including prominent politicians) have argued that in the Internet age we need only adapt the policy to include digital locks on the digital filing cabinets and company networks.  This is dangerously flawed thinking.  The complexity of accessing a piece of private information (illegitimately) has gone right down.  Copying an entire warehouse full of private data would take months (before the Internet).  Today if the files are centralised the same task can be accomplished in hours, sometimes minutes.

* It has been argued by some neuroscientists that we will not be able to read people's minds even if we can trace the firing of their neurons explicitly.  The argument goes that the inner workings of people's minds are probably unique to them (almost certainly true), the minds have many things going on at once in parallel (true) and that we will therefore not be able to come up with a sensible way of understanding what the neuronal signals mean (hmmmm).  The important question is "Can you hide what task you're doing by performing it in an incredibly idiosyncratic way even if your enemies can see everything you do".  Well, this is essentially equivalent to finding a way to hide what a program is doing on your computer from someone who can watch its every calculation.  No one has found a way to do this (its possible to make it a little harder though).  Many cryptographers think its impossible to do.  Certainly its very unlikely that the brain is set up to do this (Why would it evolve this way?).  This means it is likely that knowing exactly what neurons were firing in a brain would tell you what that brain was thinking.

I hope thats given you some indication of the value of this new area of research!

Economic benefits of AI

In this blog post I shall analyse in detail the economic effects that I would expect to accrue from the development of AIs approaching or surpassing the intelligence of humans.  I shall use the term people to refer to AIs (for simplicity).

Education: As people would be implemented on digital machines that would make the task of making copies much easier.  There would thus be little point training a million chefs to cook all over the country when you could just train up a handful and copy each hundreds of thousands of times.  The cost of education (to the current standard) will drop essentially to zero.  Specialised tasks that need to be performed very rarely will become much easier to perform.  Rather than hire out an expert to fix a bike you'd download a program into your brain and fix it yourself.

Speed of transport: As AIs are composed of information and can be transfered from one physical brain to another they will effectively be able to travel at near light-speed wherever they want (bandwidth maybe the most important limiting factor).  A huge section of todays economies is dedicated to moving people around.  When they can travel along optical fibres this cost will be near zero.  In addition as AIs will not need to have a fixed abode AIs can be highly concentrated in a few mega cities increasing the efficiency of transporting goods.  Less materials will need to be transported because the location of factories can be placed closer to where the raw materials are and where the finished products are needed (little attention needs to be paid to where people live).  Also, as has been mentioned before on this blog, the conquest of space will be much simpler once AIs can move at near light-speed to and from space stations.

Fluidity of business: As retraining people will become easier and people can move around very easily the natural barriers that make the concept of a company work will break down.  It is conceivable that without these barriers people will not work for an individual company but for any number of different groups (much like people are members of many different social circles today).  Companies may still exist as legal entities but they may become much more like overlapping circles in a Venn diagram.

Near immortality:  AIs do not need to die.  Clearly this may cause significant ethical problems.  However, there are positives to this as well.  The longer people live for the more incentive they have to plan for the future.  It makes much less sense to be irresponsible the longer you live (and hence the longer you need to live with your reputation).  This should lead to better long term economic policy (at all levels of society) and to better law and order.

Health-care: This is a really significant area of spending that would become virtually worthless with the advent of AI.  Why keep your body in tip top condition when all you need to do is move to a new one when the old one starts waring out?  Why have a single body at all when you can rent the exact body you want for any purpose you have in mind?

Lower risk: If you can back yourself up at the press of a button then normally very risky activities can be made much less risky.  If the running copy of your brain gets destroyed but you backed up 2 hours ago thats essentially the same as you forgetting the last couple of hours of your life.  This is an experience that many a party goer is well aware of and willing to risk.  When even currently very risky jobs can be brought to an acceptable level of risk new industries will open up that we couldn't previously contemplate.  Weapons of all types become much less dangerous when most people are protected through back ups of their minds.  Its probably too much to hope that this might reduce the propensity to warfare.

Extra senses: There's no reason that AIs have to be limited to the 5 senses that we have.  Novel senses could be designed and developed.  Seeing in ultraviolet, infrared, sonar, radar and all manner of wave forms will become routine.  But in addition the visualisation of abstract mathematical data structures will become much easier.  Imagine the mathematics that could be performed by someone raised to see and understand four dimensions.  Difficult problems in any field (not just mathematics) could be solved by evolving an intelligence with the right set of senses are experiences to understand them.

Bureaucracy: Highly complex organisations will become possible as the amount of intelligence available to oversee a given physical task becomes greater and greater.  Systems that previously strained with bureaucracy will become easy to manage.

The economy that would emerge from the advent of AIs would be so much more efficient in so many different ways that the development of AI has got to be the most economically advantageous thing that is within our reach this century.  As I have previously observed there are terribly complex and disruptive problems that would also emerge with AIs (even if you ignore the possibility of conflict between humans ans AIs).  It seems to me though that such an AI society is by far the closest thing, within the reach of modern society, to a spiritual realm.

Order theory and a theological argument

This blog post is going to be about a very particular style of argument that I have heard from the mouth of a Buddhist and I have read in C.S.Lewis's "Mere Christianity".  I find the arguments interesting because they are essentially mathematical in nature.  This makes them amenable to the rigorous treatment of mathematical logic.

Firstly a rough template of the arguments:

* A thing may be x like.

* Some things are more x like than other things.

* For this to be the case there must be a perfectly x like thing (call it X) which the other things are judged against.

* If X is perfectly x like then for any Y we must have that Y is less x like than X or that Y is in fact X.

This argument has been used with morality.  Then the things in discussion are moral actions and they can be good or bad to differing extents.  This argument can be used with minds which can be more or less knowledgeable than each other (or more or less enlightened than each other).  The argument can be used with entities which can be more or less powerful than each other.

In each of these cases the arguments are meant to conclude respectively that there is a universal morality, that there is a state of universal enlightenment and that there is a unique all powerful entity (which I guess you hope is also good).

There are many problems with these arguments including:

A) The arguments need to show that something exists (or could exist in the case of enlightenment) in the world NOT as a mathematical abstraction (with the exception of the case of the universal morality).  The argument above is an argument for mathematical existence not physical existence.

B) It simply is not the case that whenever you have many different options some of which can be ranked higher than each other that you must have a unique best option.

Here is a simple game called "My daddy's richer than your daddy!".  Two players write down a number privately on a piece of paper.  They show the numbers to each other.  Whoever has the highest number wins.  It is always better to write a higher number down but there is no highest number that can be written down.  A religious person might object by inventing quantities greater than normal numbers.  A mathematician cannot complain because that is essentially what mathematicians call ordinals.  However, then we get the Burali-Forti paradox that there is no highest ordinal (I'm simplifying here for clarity) and we are left with a cast iron instance of this argument style failing.  Notice that what we really need rather than a top number is a concept of quantity.  The concept of quantity is not itself a number.  In the same way to judge morality we need an ethical theory not a perfectly moral entity.

There are further problems including the problem that one might not be able to compare which of two things is more x like.

I think, however that what is really going on here is that people are leaving out assumptions that would make the arguments stand up properly.  I shall give the simplest assumptions I can think of that would suffice to make the above argument work and then we can look at how appropriate they are in the above cases.

1) If we have a set of things that are x like there is something that is at least as x like as all of them.

2) There is a sensible way of ordering things as to how x like they are in the first place.

3) The quantity of objects we are considering is bounded (but possibly infinite).

The problem for this style of argument is that (1) and (2) are generally just as contentious as the results the arguments seek to prove.  (3) is never really going to be true when the argument seeks to prove something interesting.  In all the cases above (1) and (2) are contentious and (3) is false.

It is my opinion that there are other theological arguments that we would do well to examine from the perspective of mathematical logic.

Rules of thumb for vegetarian ethics

Once one has decided that animals require moral consideration the question "How should you weigh animal's needs against each other (and us)?" arises.  In this post I shall give some of my thinking on the issue and finish with a list of thorny questions to tax the reader.  By person I shall mean any creature with neurons.

I am going to work within my libertarian framework here (see the last post on libertarianism).  The libertarian position still requires a base ethical theory to supply answers to various questions and thats what I discuss here.  For simplicity a utilitarian base theory is used as most of the time such a base theory works quite well (I think there are situations where utilitarianism is lacking particularly with regard to ethical situations regarding AIs)

The rules of thumb that I propose are only meant for the following two sorts of situations.  In other situations things are more complicated.

1) Suppose you have a choice of several courses of action.  Each of these courses of action will harm someone who has not consented.  Let us also assume that there are no positives.  Which should be chosen?

2) Now suppose that each of the courses of action confers a positive benefit on some individual or individuals.  No one objects to your courses of action.  Which should be chosen?

Firstly within a given species (ignoring sexual differences/type differences that do occur in some other species) I use a rule of thumb I call the quality of life year.  You multiply the quality of life gained/lost by the number of years it is gained/lost for to arrive at a measure of how beneficial/detrimental the intervention is.  This (I am amazed to discover) is the way the national health service of the UK decides on health care decisions.  I heartily approve!  It is difficult to decide what value to give different types of effect but the NHS has developed an interesting variety of methods for this sort of issue.

Secondly it will occasionally be necessary to decide how to weigh different animals (including us) against each other.  We shall now assume that all other things are equal and that it is purely a decision based on species.

I argue in this case that a good rule of thumb is to weigh the animal according to the number of active synapses in the neural system of the creatures in question (this does imply some animals are worth as much or more than people).  Here I shall give a brief thought experiment to support this:

Suppose that there is a world in which blobs of brain move around like polyps.  Sometimes many small brains come together into a larger unit to do some serious thinking.  The brains then rearrange their synapses to connect with each other and become essentially one super brain.  Now notice that it would be practically impossible to ascertain to what degree the brains are engaged in meaningful communication.  Its also difficult to see from the outside what the contents of the mind of those brains are (and probably even more difficult (not to mention dubious) to use this knowledge in ethical decisions).  As such we must observe that when 100 human sized brains come together to make a gigantic brain that we can't tell whether to treat them as 100 human sized brains or one gigantic brain.  In such a situation then the only reasonable and applicable morality is to give the brains ethical consideration according to the number of synapses they possess.  Then the gigantic brain will be treated like 100 human sized brains whether or not its acting as 100 independent agents or 1 super agent.  Now we can observe that the assumptions made here apply to animal's brains as surely as to the fictional blob brains.  It is also a potential ethic to apply when we create AIs (although possibly problematic).

Finally we must decide how to weigh creatures when their brains change size over time.  I addressed this issue partially in my post on abortion.  If a being has X many synapses and Y many years of expected life left then the loss of that being's life produces a cost of XY irrespective of the size the beings brain may eventually attain.  The loss of potential cannot be regarded as being any loss at all unless the potential is unique (i.e. killing the last mating pair of a species at their birth).  As such only the current brain size should be counted.  This reasoning results in a compatibility between a pro abortion stance and a vegetarian one.

Further questions:

1) If I could make a perfect copy of your brain/mind would it be murder to destroy that copy an instant after creating it?

2) Is it really the case that 1 billion ants should be given 1 billion times more moral weight than 1 ant.

3) What is the value of uniqueness in the mental realm?

4) How does all this relate to issues surrounding AIs?

I'm not going to put my views on these questions yet but I might address them by email or in further posts....

The problem with probability

Today I am going to spill the beans about a huge gap in our understanding of the world and of mathematics.  In my view this is probably the biggest stumbling block in our understanding of the philosophy of science and a massive obstruction in the way of advancing mathematics.  That stumbling block is probability.

I recently read an article in a lovely little book entitled "What we believe but cannot prove" (ed. John Brockman).  The article suggested a simple problem in mathematics that the author (Freeman Dyson) believes we will never resolve.  The problem is the following:  Consider numbers in the series 1,2,4,8,16,32 (where each time you multiply by 2) and in the series 1,5,25,625 (where each time you multiply by 5).  Can you obtain a number in the second series by reversing the order of a number for the first series (i.e 1,2,4,8,61,23)? 

Dyson argues that (upon the assumption that the digits in the first series behave kind of randomly) that the answer is probably no (With less than 0.001% chance) but that this must also be unprovable if the assumption is correct.  It seems a plausable argument but it got me thinking because we mathematicians do not have a model of probability.

Certainly there are probability theorists and statisticians but they are just doing the best they can of a bad job.  I'll do my best to explain the extent of the problem in terms not requiring a mathematical education.

First, there is an assumption that is necessary.  A few mavericks do not agree with the assumption but the majority of mathematicians agree with it and use it every day.

If you have a set of sets where each one has something in it and no two share an member then:  There is another set containing exactly one thing from each of those sets.   I'm hoping that seems obvious to you.  Its called the axiom of choice because it allows you to make lots of choices all at once.

Probability theory works quite well when we have a finite number of things that are random.  It only gets into trouble when you do something like flip a coin infinitely many times.  This matters because in the philosophy of science this sort of thing is going on all the time.  It also matters in mathematics because damn near everything we do these days can be linked to such possibilities.

Now suppose I promise to give you £100 if the sequence of coin tosses has a certain property X.  How much should you pay to take up the bet?  If I said the coin sequence must contain a head, then you'd bet anything up to £100.  If I said that the coin sequence must never contain a string of heads longer than any string of tails before it then you shouldn't bet (this is almost impossible).  You would be justified in wanting a cut and dried betting strategy for any property X so long as it's properly defined (no abiguity is allowed).  Here's the rub:  Assuming the axiom of choice there is a property such that there is no sensible amount of money to bet.  That doesn't mean you should bet £0 or £100. It means whatever money you'd bet you can't justify it.  Standard probability theory cannot deal with these situations.  This is the main reason some mathematicians doubt the axiom of choice.  But this axiom does really seem obvious.

I can't tell you what the problem really is or how it should be resolved.  But I can give you my suspicions.  I think our notion of probability doesn't work with our philosophy of science.  I think the notion of infinitely many random events is the notion causing trouble.  I shall write another post explaining why I think Occham's razor sheds light on this (and taking my old post further at the same time).