The false premise in Moore’s open question argument

Normative statements, such as “killing is bad” or “thou shalt not kill” do not follow logically from merely informative facts.  Normative statements are prescriptive, in the sense that they tell you what you should do.   In contrast, informative facts or descriptive statements, also known as positive facts, merely describe a situation without offering a prescription about what to do.  For instance, the statement “people usually dislike being killed”, taken at face value, is not prescriptive (or normative), but rather simply a description of something that people dislike.  It has been known since the time of David Hume that there is a “gap” when one attempts to logically deduce a normative fact from a merely descriptive fact. In the above example, no amount of logical acrobatics will get you from the premise “people usually dislike being killed”  to the conclusion “thou shalt not kill.” This is the famous is-ought problem posed by David Hume, also known as Hume’s guillotine.

When I was in school, my father gave me a paperback with selected texts on philosophy, which included some by Hume.  I was mesmerized by the discussions concerning empirical knowledge.   Of all of the ideas and contributions that can be traced back to Hume, one idea has kept torturing me over many years: the dualism of facts and decisions implied by Hume’s guillotine.    This dualism in principle prevents us from deriving normative statements — hence ethics — using the scientific method, because the latter deals only with descriptive statements (see below).

Years later, in my 30s, I eagerly read  Karl Popper, who no doubt borrowed many of his ideas from Hume. For instance, according to Popper,  there is no certain empirical knowledge. All certain knowledge must be a priori and all empirical knowledge is tinged with uncertainty.   In The Open Societies and its Enemies, Popper explains very clearly that any descriptive fact can be used to argue a decision in both ways.  For example, if the fact of an almighty god’s existence could be somehow scientifically proven, this fact does not automatically imply that we ought to follow this god.   We could instead try to argue that this god must be dethroned.   Similarly, if a medicine fails to have a positive effect on a patient, one could argue that the medicine should be stopped. But one could also argue that we should double or triple the dosage.   Hume’s guillotine cuts both ways.

If you assume that Hume’s guillotine is a valid way of reasoning, then there are limits to how much the scientific method can be used to improve our decision making.   This limitation is due to how science essentially generates facts, whereas to make a decision one requires a value.  Science is, in this view, essentially anormative or amoral.

I thus read with great interest some recent texts by Sam Harris in which he argues that Hume’s guillotine is, after all is said and done, a mistake!  I was, to say the least, quite shocked.  Harris does not address the philosophical literature and others have pointed out that he essentially ignores centuries of philosophical thought in simply dismissing Hume’s guillotine.

But I still asked myself: could Hume’s guillotine be a mistake after all?

In this context, I decided to revisit one of the main pillars that support Hume’s guillotine, which is known as the open question argument, originally due to G. E. Moore in 1903. Moore argues logically that “good”  cannot be a natural property, for the following reason:

Premise 1: If a natural property X is identical to “good”, then the question “Is X identical to good?”  is a meaningless tautology.

Premise 2: The question “Is X identical to good?” is not meaningless and is a genuinely meaningful “open question”.

We thus get a contradiction. Hence, X cannot be identical to “good”.

The open question argument makes it very difficult to try to define “good” in naturalistic terms.   This is turn makes it very difficult to address “good” and “bad”  using the scientific method. (A notable exception is the “science” of medicine, where disease is unquestionably seen as bad and recovery is considered good.)

Below, I argue that Premise 1 is wrong.  The points raised here are not new, however, the particular example that I give  below probably has never been used in the context of the open question argument.

Consider the following question:

Is the number 1 identical to  \sqrt[3]{8 + 3 \sqrt{21}} + \sqrt[3]{8 - 3 \sqrt{21}} ?

To most readers, the answer to this question is not at all obvious.  Hence,  there is no sense in which this question is meaningless.   Moreover, the question is a simple binary yes/no question with a clear answer, but this answer is not immediately clear unless one tries to solve the maths.

It turns out that the answer to this question is YES, which means that we have a tautology:

1=\sqrt[3]{8 + 3 \sqrt{21}} + \sqrt[3]{8 - 3 \sqrt{21}}   .

Premise 1 of the open question argument assumes that tautological questions are meaningless, but we have already seen that this particular tautological question is not meaningless.

The above example has convinced me that the open question argument is based on a false premise and is therefore not valid.

More and more, I have become convinced that Hume (and Popper and Moore etc.) were wrong.   Perhaps  facts are not so ethically (and aesthetically) neutral after all.  The example of medicine comes to mind. If medicine can use the scientific method to improve our physical and mental health, I don’t see how it is logically possible for ethics to remain beyond the reach of the scientific method.

Many years before, I came across the same sentiment in the writings of Abraham Maslow, who also greatly influenced me (I remember how exciting it was for me to first read his list of B-values).  Now,  after reading Sam Harris, I am more than ever convinced that the scientific method will still deliver many new and priceless jewels of knowledge.

Given the recent populist wave in politics that seeks to discredit science, such views may thus come as a surprise, or at least as overly optimistic and naïve.   But I stand by my views.

I believe that we are at the dawn of an exciting new era where science will contribute significantly towards improving the moral and ethical dimensions of human social life and of our interactions with the environment.


I thank Marcos G. E. da Luz for the  mathematical example involving cube roots. I thank Helcio Felippe Jr. for discussions and feedback.

Perfect matchings of a complete graph and the double factorial

I have a longstanding interest in graph theory applied to statistical mechanics. For example, a few years ago I published a paper in Physical Review E that explores the correspondence between spanning trees and the Ising model on the square lattice (see PDF here). More recently, I became fascinated by perfect matchings.

There is a deep relationship between perfect matchings in graph theory on the one hand and the theory of equilibrium statistical mechanics on the other hand. Perfect matchings are relevant to dimer models and to certain lattice spin models, for instance.

As entertainment and also as a birthday present to myself for my 50th anniversary this week, I decided to figure out for myself exactly why it is that the double factorial appears in the expression for the number {K_{2n}} of perfect matchings of complete graphs of {2n} vertices. The number {K_{2n}} counts the number of different ways of pairing all {2n} vertices with exactly {n} edges, without any vertex being left out or having more than one connecting edge. It turns out that

\displaystyle K_{2n} = (2n-1)!! \ \ \ \ \ (1)

where

\displaystyle (2n-1)!!= {(2n)! \over n!2^n} ~. \ \ \ \ \ (2)

It is important not to confuse the double factorial with the factorial applied twice,

\displaystyle N!! \neq (N!)! ~~, \ \ \ \ \ (3)

rather the double factorial is defined as

\displaystyle {\displaystyle N!!=\prod _{k=0}^{\left\lceil {\frac {N}{2}}\right\rceil -1}(N-2k)} \ \ \ \ \ (4)

where the ceiling function {\lceil x \rceil} gives the smallest integer not smaller than {x}. In other words,

\displaystyle N!! = \left\{ \begin{array}{ll} N(N-2)(N-4)\ldots (2), & \quad {\rm for~ even}~N \\ N(N-2) (N-4)\ldots (1), & \quad{\rm for~odd}  ~N ~~~ .\\ \end{array} \right.  \ \ \ \  (5)

Let us first calculate the number of perfect matchings. Given {2n} vertices, we can pair the 1st vertex with {2n-1} others with a single edge. Once the first edge is assigned, there are {2n-2= 2(n-1)} vertices left that need to be paired. So

\displaystyle K_{2n} = (2n-1) K_{2(n-1)} ~~. \ \ \ \ \ (6)

We can iterate this recursion relation:

\displaystyle \begin{array}{rcl} K_{2n}&=& (2n-1)(2(n-1)-1) (2(n-2)-1) \ldots (5)(3)K_{2} \\ &=& (2n-1)(2n-3)(2n-5)\ldots (1)\\ &=& (2n-1)!! ~~. \end{array}

So this is why the double factorial appears in (1).

To arrive at (2), we proceed to express the double factorial in terms of the standard factorial. First, notice that {2^m m!} can be written as

\displaystyle 2^m m! = (2m)(2m-2)\ldots 2 = (2m)!!, \ \ \ \ \ (7)

so that

\displaystyle (2m)!! = 2^m m! \ \ \ \ \ (8)

For the double factorial with odd arguments {2m - 1} we proceed as follows. Note that the double factorial contains either only odd or only even factors, which can can combine to obtain the standard factorial thus:

\displaystyle n!= n!! (n-1)!! ~~. \ \ \ \ \ (9)

This expression relates the double factorial of odd and even arguments. Let {n=2m}. Then the above gives us

\displaystyle (2m-1)!! = {(2m)! \over (2m)!!} = {(2m)! \over 2^m m!} \ \ \ \ \ (10)

from which follows (2).

 

Internal energy of the 2D Ising model

The above domain coloring plot shows an analytic continuation to the complex plane of the internal energy of the 2D Ising model. The domain variable is z=\tanh J \beta, which is a standard high temperature variable used to study the 2D Ising model.

I originally made the plot in 2014 soon after I first began to study the exact solution of the Ising model. A few days ago, I happened to find this old picture from 2014 and decided to share it here.

What a beautiful model! I had forgotten just how beautiful the structure is.

The 2 circles represent thermodynamic limit of the zeroes of the partition function. In this limit, singularities appear where the zeroes would be, as can be seen from Onsager’s formula. The critical temperature of the Ising model corresponds to the singularities on the real line, of which there are 4. The two singularities closer to the origin on the real line give the well known positive and negative critical temperatures \pm \beta_c J \approx 0.44. The other two give physically non-realizable complex critical temperatures, which are related to the standard critical temperatures via a symmetry known as the inversion relation (see MT Jaekel and JM Maillard, Symmetry relations in exactly soluble models, Journal of Physics A: Mathematical and General, 1982).


Mathematica Code:

f[z_] := (-\[Pi] (1 + z^2)^2 +
2 (1 – 6 z^2 + z^4) EllipticK[(16 z^2 (-1 + z^2)^2)/(1 + z^2)^4])/(
2 \[Pi] (z + z^3));

paint[z_] :=
Module[{x = Re[z], y = Im[z]},
color = Hue[Rescale[ArcTan[-x, -y], {-Pi, Pi}]];
shade = Mod[Log[2, Abs[x + I y]], 1];
Darker[color, shade/4]];

ParametricPlot[{x, y}, {x, -3, 3}, {y, -3, 3},
ColorFunctionScaling -> False,
ColorFunction -> Function[{x, y}, paint[f[x + y I]]], Frame -> True,
MaxRecursion -> 1, PlotPoints -> 300, PlotRangePadding -> 0,
Axes -> False , Mesh -> 29 , Mesh -> {Range[-5, 5], Range[-5, 5]},
MeshFunctions -> {(Re@f[#1 + I #2] &), (Im@f[#1 + I #2] &)},
PlotRangePadding -> 0, MeshStyle -> Opacity[0.3], ImageSize -> 400]

Artigo científico mais citado da UFRN e da UFAL completa 20 anos

O artigo mais citado da UFRN e da UFAL completou 20 anos em 2019 .  Estou, obviamente, desconsiderando artigos envolvendo grandes colaborações (vide abaixo).

A Web of Science produz o seguinte, procurando pelo endereço com as palavras “rio”, “grande” e “norte” e ordenando os resultados pelo número de citações:

(Para maior clareza, veja a versão PDF neste link).

O artigo mais citado, do ano 1999 e publicado na Nature,  pode ser encontrado neste link aqui.  O PDF completo deste e dos meus outros artigos podem ser encontrados aqui.

O levantamento acima desconsidera  artigos envolvendo grandes colaborações, tais como os da LIGO e da SDSS. Em geral, na comunidade científica não se faz comparação direta entre artigos com poucos autores e artigos de grandes colaborações, que podem ter centenas de autores. Levando em consideração grandes colaborações, 3 artigos são mais citados do que o artigo de 1999 (veja aqui as citações).  Dois são ligados à LIGO e o outro à SDSS.

Aparecem nos meus endereços tanto a afiliação institucional da UFRN como também da UFAL.  Procurando na Web of Science pelo endereço com as palavras “univ”, “fed” e “alagoas” encontramos o seguinte:

(Para maior clareza, veja a versão PDF neste link)

Portanto, o artigo de 1999, que completou 20 anos em 2019, é o mais citado da UFRN como também da UFAL.

Para comemorar a ocasião, a  Agecom/UFRN publicou uma matéria sobre o assunto, veja neste link.

 

Podcast sobre Ayahuasca

Aqui está o link de um podcast onde falo um pouco sobre Ayahuasca.

 

O podcast, da Fronteiras em Ciência, saiu em Abril de 2018. O artigo original na Scientific Reports está aqui. Essa pesquisa fez parte da tese de doutorado de Aline Viol, sob minha orientação.