Science and Mathematics

Mathematics deals with perfection, with absolute truths. Scientific theories on the other hand always have a sense of approximation. Whereas a mathematical statement like the theorem of Pythagoras holds true for all right angled triangles (and hence may be used to define a right angled triangle), scientific theories like the theory of gravitation are subject to constant revision and updating. In science, people come up with hypotheses to explain phenomena. If experiments then corroborate the predictions made by these hypotheses, then they gain credibility and are elevated to the status of a theory and become part of our everyday understanding of the world around us.

Here’s a question which serves to illustrate the difference. You have a chessboard in which two squares from opposite corners are removed. So instead of 64 squares, you now have 62. And you are given 31 dominoes that cover 2 squares each. Is it possible to cover the board with the dominoes? (source : Simon Singh, Fermat’s Last Theorem)

The scientific way to answer this question would be to try filling the board in different ways and see for yourself whether the question permits a solution. After a few dozen attempts you may come to the conclusion that it is not possible to fill the squares with dominoes but still an element of doubt prevails. This is now a theory based on experiment and is readily overturned if even a single counter-example can be produced.

Another way is to argue using logic. You notice that the squares on opposite corners of a chessboard are always of the same colour (say, black). Hence, if you remove those squares you’ll be left with 32 white squares and 30 black squares. Also any two adjacent squares in a chessboard (which we wish to cover using dominoes) must necessarily be of opposite colours. Hence after 30 dominoes, we’ll be left with two white squares and a single domino. But since the two white squares cannot be adjacent, we can conclude with absolute certainty that  it is impossible to fill the squares with the dominoes. It is this absolute nature that gives mathematics its beauty.

Prof. V. Balakrishnan once told us in class that although we (physicists) know that the photon has zero rest mass, the state-of-the-art experiments can only confirm that its mass is less than 10-54 kg. So if the photon has a mass, it must be less than 10-54 kg. This blatant acceptance of its limitations, I believe,  may be why we  would never be able to convince a creationist (or a climate-change denier) of the truth and explanatory power of some of the theories in science. They just don’t understand the way science works1, that there is such a thing called the relativity of wrong. As Asimov nicely puts it:

When people thought the earth was flat, they were wrong. When people thought the earth was spherical, they were wrong. But if you think that thinking the earth is spherical is just as wrong as thinking the earth is flat, then your view is wronger than both of them put together.

The shape of the Earth is not oblate spheroidal either though it is a better approximation still. Now the shape of the Earth has a special name – geoid – which in Greek means “shaped like the Earth” (The mathematics now gets more complicated though.) So our  Earth is a geoid (no surprises there). To me, it is this sense of adventure and exploration, deepening our knowledge  while at the same time remaining humble of its limitations that gives science its beauty.


1. There is this funny little story about the philosopher Ludwig Wittgenstein (1889-1951). Once he asked one of his friends why people always say that it was natural for men to assume that the sun went around the earth rather than the earth was rotating. His friend said: “Well, obviously, because it just looks as if the sun is going around the earth.” To which the philosopher replied: “Well, what would it look like if it had looked as if the earth were rotating?”

Are you my friend?

Are you my friend, my friend?
If so then please do reveal
Whether you feel as I feel
That it sometimes is all just a means to an end.

Isn’t true friendship as Platonic a notion as a perfect triangle?
So feeble, fickle and frivolous
Fed to minds ever so credulous
Who but then find themselves in a tangle

Why don’t you make friends and see you say
Write poems I cannot, but I do try
Make friends I dare not, for I may cry
At least I’m not so unhappy this way.

All the world’s a stage and we all are players
Making friends and forgetting them so fast
Today’s friends are tomorrow’s past
In our constant search for new foyers.

Where can I find a friend, where?
Who’ll my foolish notions upend
You might say that you’ll, my friend
But you’re not there when you’re not there.

Random thoughts

The German word for disappointment is “Enttäuschung which when literally translated means disillusionment/disenchantment. That really makes you think. You need to be under some kind of an illusion – a false sense that everything will turn out as you had expected, or people will understand you and act as you want them to – so as to feel a sense of disappointment.  When you are disappointed, you rid yourself of that illusion. Germans sure know what they are talking about.

Speaking of disappointment, I see some people use “anyways”. I don’t know why. It is an adverb and adverbs are never plural. And “anyway” and “any way” mean two different things. It is just like “you’re” and “your”. I was telling my friend the other day that the printer can’t print red because it can only print primary colours. And he was puzzled. Isn’t red a primary colour? No, that’s for light where the principle of addition applies (RGB). For paints, the primary colours are cyan, yellow, and magenta (ok and black). Did you notice that I slipped in an Oxford comma (Harvard comma if you’re an American) in the previous sentence? Is that necessary? I don’t know but it looks weird to me now. Weird would be a weird word for Germans with their “Wien und der Wein”. Coming to think of it it is weird in English too as it boldly neglects the “i before e, except after c” rule .

So where was I? Paints and dyes work on the principle of subtraction (CMYK system), I said. He thought that you’d get white when you mix red, green and blue paints. Or maybe (or maybe “may be”? Nah, I may be stretching it there.) he didn’t think  and just wanted to say something which I  feel is even worse. Anyway, what’s the term again, ah “Enttäuschung”.

I’m going home today, only to come back in a few days for the coveted degree. Getting into IIT is a pain they say. But getting out, oh boy, is in a different league of difficulty. And I need to get out so that I can get back in to do a PhD. AIl that wasted effort. But there is hope. I’ll have the 14-idly-ghee-sambar at Saravana Bhavan for dinner (at the railway station). Yeah the dish is called just that and no a zoologist didn’t come up with it (if you get the reference). Home sweet home.

The theory of intelligent falling

“Do you know every creationist has a blind spot?”

– Randolph Nesse

Evolution is only a theory. Yeah, so is gravity. And germ theory. But you don’t see people carrying banners that say “Teach the controversy” for those theories. Teach the controversy! There is no controversy. All the scientists with any level of credibility and knowledge on the matter accept evolution as a fact. The disputes are only between scientists and non-scientists, and since the topic is science, the non-scientists don’t get to vote.

But the gaps in the fossil records. Show me the transitional fossils. Well, every time a scientist comes up with one, it only makes the creationists even more happy as there are now two new gaps to explain in the fossil record. Intelligent design you say. Hmm, I bet I know who your intelligent designer is going to be.  How about an intelligent theory of falling. Then you can teach the controversy in the theory of gravitation. As Nick Matzke rightly quipped :

Intelligent design is creationism in a cheap tuxedo.

 

More evidence? First get hold a good book on evolution. I recommend The Greatest Show on Earth by Richard Dawkins. He has poured his heart out in explaining the various evidence available for evolution, not just from fossil records, but from radiometric dating, palaeontology, embryology, anatomy, genetics, artificial breeding and geography. The following is an except from the book (Chapter 11, pp.356) which shows an example of unintelligent design in mammals (that includes us).

 

[Recurrent laryngeal nerve] is a branch of one of the cranial nerves, the vagus (the name means ‘wandering’ and is apt), has various branches, two of which go to the heart, and two on each side to the larynx (voice box in mammals). On each side of the neck, one of the branches of the laryngeal nerve goes straight to the larynx, following a direct route such as a designer might have chosen. The other one goes to the larynx via an astonishing detour. It dives right down into the chest, loops around one of the main arteries leaving the heart (a different artery on the left and right sides, but the principle is the same), and then heads back up the neck to its destination. If you think of it as a product of design, the recurrent laryngeal nerve is a disgrace.

The following video shows the recurrent laryngeal nerve in a giraffe. As you can see the detour it takes from the brain to reach the larynx suggests anything but intelligent design. Viewer discretion is advised  as the video contains graphic images.

No engineer would ever make a mistake like that. If you are still not convinced, you can just float the hell off.

Casting out 9s

Most of us know the rule to check whether a number is divisible by 9. You sum the digits of the number and if the result is a multiple of 9, then the number itself is a multiple of 9. For example, 4572 is a multiple of 9 as 4 + 5 + 7 + 2 = 18 (or if you want to go further, 1 + 8 = 9) which is a multiple of 9. But not many of us know why this is so. Why should the numbers sum to a multiple of 9 (or to 9 if you keep summing till you end up with one digit)?

Well, it is because we have a decimal (base 10) number system. Take any general number, say 3587. We have 3587 = 3×1000 + 5×100 + 8×10 + 7×1, and each of the elements in the set {1, 10, 100, 1000,…} is one away from a multiple of 9 {0, 9, 99, 999,…}. This means 3587 is 5 (3 + 5 + 8 + 7 = 23, 2 + 3 = 5) away from a multiple of 9. This in essence is modulo 9 arithmetic; summing the digits gives us the remainder when a number is divided by 9. The same trick works for 3 insofar as checking for divisibility. Casting out 9s is useful to check the correctness of your arithmetic (all 4 basic operations). More on that here.

Can you extend the same logic to check why the following rule holds for divisibility by 11? (A number is divisible by 11 only if you arrive at 0 or 11 while alternately adding and subtracting the digits of the number, eg. 25476 is divisible by 11 as 2 – 5 + 4 – 7 + 6 = 0)

Cognitive dissonance

“Extraordinary claims require extraordinary evidence”

– Carl Sagan (1934 – 1996)

These are excerpts of a discussion I had with one Mr. Kroor one fine evening when we had nothing better to do. We were debating over the talk Sam Harris gave at TED. This is my take on the talk.

The central theses of Sam Harris’ talk were:
(i) There are right and wrong answers to moral questions, and
(ii) Science can objectively tell us what is right or wrong.

Claim (i) betrays a stance of moral absolutism/naturalism. When one makes such sweeping statements, one should substantiate it with adequate evidence. Else they just remain claims rather than facts. Also care must be taken so as to not fall prey to Hume’s Guillotine (the is/ought problem, or I/O for short). Consider the following statements.

(i) Punching people in the face causes them pain.
(ii) One ought not cause people pain.
(iii) One ought not punch people in the face.

When you take a stance on a moral issue (like I do not want to cause people pain), you are being normative (read subjective). You have taken your stance based on what you as an individual think is right or wrong. In other words you are biased. Once you have taken that stance you look for facts; facts that tell you what causes pain to people (punching people in the face for instance). Notice that this is a factual statement and not a normative one. And this is where science comes in handy.

Finally, armed with scientific facts and the power of reasoning, one goes about their life doing what they feel is the right thing to do for any situation (stop punching people in the face because causing people pain is wrong or you may get punched back which will cause you a lot of pain). So the role of science is only as an arbiter in

(i) deciding the facts that carry the required import, and
(ii) checking for logical inconsistencies.

Harris says in his talk:

Values are a certain kind of fact. They are facts about the well-being of conscious creatures.

In the example above we saw that the factual statement (i) was distinct from the statement on moral values (ii). One cannot directly go from statement (i) to statement (iii) without invoking some form of statement (ii) and one can keep on adding similar normative statements to plug any leaks. And these statements are reflective of our biases and does not follow from (i). In other words, neither statement (ii) nor (iii) is a logical conclusion that one can derive from a factual statement. So a value statement like (ii) or (iii) are not facts in themselves.

In conclusion, facts on values or value judgements (like statement (i)) are not values themselves. And hence, science cannot give us right or wrong answers to moral questions since it deals with facts and factual claims. Science can give us facts about wellbeing of creatures. Right or wrong answers to moral questions differ from person to person (from claim to claim). These value statements which carry within them subjective biases hence cannot logically follow from factual statements (i.e., they cannot themselves be factual statements).

Kroor’s reply

Consider the following:

(i) I don’t like pain [Universally observed in animals. Humans, say, presently]
(ii) If I cause someone pain, they will retaliate [Almost universal in animals; hence in humans]
(iii) Therefore, I should not pain someone else. [Evidently normative prescription; arrived at from a universal observation and an almost universal observation generalized slightly].

Is this not what Harris is saying when he says “Values are a certain kind of fact. They are facts about the wellbeing of conscious creatures”? If yes, we’ve apparently proved him right. If not, you take objection to the golden rule or resort to the possibility that I’m a masochist. I think that (i) and (ii) hold quite generally.

Can moral statements have absolute answers?

I don’t know.

Can most of conventionally accepted morality be derived from (i) and (ii) above?

I think so.

So, can reason/logic give answers to moral questions, given (i) and (ii) above?

Evidently.

Have I shown that all morality is derivable inductively from the golden rule?

No. Abortion, for example. Are you hurting the foetus when you abort it? We don’t know. Will the foetus hurt you back, even if you are? Definitely not. Is abortion, therefore, murder? I don’t know.

A rebuttal

(i) I don’t like pain [Universally observed in animals. Humans, say, presently]

(ii)If I cause someone pain, they will retaliate [Almost universal in animals; hence in humans]

(iii)Therefore, I should not pain someone else. [Evidently normative prescription; arrived at from a universal observation and an almost universal observation generalized slightly].

Claims (i) and (ii) are normative and hence are ought claims. To say that animals (in this case you (we’re the fifth ape)) don’t like pain is an opinion (subjective) whereas a statement like pain causes suffering in animals is objective and factual. To say animals will/should retaliate when subjected to pain is again your opinion. Facts may tell you that very often they do. It does not mean that they always have to do which is a hasty generalization on your part. In short you have not given me any factual claims to consider your argument seriously.

Let me rephrase your argument for clarity.

(F1) Pain causes suffering in animals (fact).
(C)  I should not pain animals (conclusion).

As I said earlier, to reach the conclusion given above you need to make a subjective claim like (S1) I don’t like paining animals. This can indeed be reinforced by another subjective claim (animals are cute or torture is bad or I am a Jain etc). If you then now add another observed fact that (F2) animals often retaliate when tortured you need another subjective claim (S2) animals will torture me when I pain them. The two subjective statements can then be combined to read (S) I don’t like paining animals because they will retaliate if I pain them. There is simply no way to work around this problem. The subjective claim simply does not follow from facts.


Is this not what Harris is saying when he says “Values are a certain kind of fact. They are facts about the well-being of conscious creatures”? If yes, we’ve apparently proved him right. If not, you take objection to the golden rule or resort to the possibility that I’m a masochist. I think that (i) and (ii) hold quite generally.

I hope this is not what he meant. If yes then he has made the same mistake as you in not understanding the precepts of moral relativism and the I/O distinction. There is a reason why I/O problem has stood the test of time. But I think he’s too smart to make such a fundamental error. Waiting for further clarifications from his book.

Can moral statements have absolute answers?
I don’t know.

No as per current understanding.

Can most of conventionally accepted morality be derived from (i) and (ii)above?
I think so.

Circular reasoning. From the statements “Bible is the word of God” (normative) and the “Bible says that God exists” (factual) you cannot reach a conclusion that “God exists”. This is a classic example of begging the question. In other words the normative statement does not follow from the factual claim. I don’t know how many times I’ve had to repeat this.

So, can reason/logic give answers to moral questions, given (i) and (ii)  above?
Evidently.

Evidently no (see above).

Have I shown that all morality is derivable inductively from the golden rule?
No. Abortion, for example. Are you hurting the foetus when you abort it? We don’t know. Will  the foetus hurt you back, even if you are? Definitely not. Is abortion, therefore,  murder? I don’t know.

Golden rule is not a fact. It’s a choice (which you either choose to follow or not). Other than that I find no flaws there. Nice reductio ad absurdum to your own argument there! (see question above)



Epilogue

While I personally don’t condone immoral deeds, I definitely do object to the statement that science can be used to “derive” answers to moral questions. There are a lot of grey areas and the potential for abuse if such systems get implemented needs no mention.

Science, Religion and Morality – Part 2.71828182845904523536….

Sam Harris recently gave a talk at TED on “Science and Morality”. Having stirred up a hornet’s nest with his audacious thesis ‘Science can answer moral questions’, he discusses how scientific precepts may be used to logically conclude whether certain actions are morally right or wrong.

I would surely post on my take on the matter real soon. For those of you who don’t quite like the reductionist attitude of Sam (that was a big giveaway) here’s a brilliant rebuttal by Sean Carrol.