The following selection paraphrased from the article on ‘Morality and Evolutionary Biology’ from Stanford Encyclopedia of Philosophy is an objection to metaphysical naturalism, but it is also relevant to ethics as well.

# The Challenge from Irreducible Pluralism

“Lets take mathematics and evolution as an example. If we say mathematical proposition x (e.g. ‘There is always a prime number between the integer n and 2n.’ ), we can determine the truth of this proposition by use of mathematical reasoning. We do not say x is true because there is an evolutionary advantage to x being true. A moral realist would argue that moral philosophy is similar to mathematics ( or physics, chemistry, etc). For example the statement that interracial marriages is wrong one can use moral reason to say this statement is false in light of the moral truth that all human beings have the same moral worth. In the same way we can use autonomous mathematical reason to evaluate mathematics we can use autonomous moral reason to evaluate ethics. Of course moral reasoning can be influenced by culture and biological factors and there are other (would argue less) plausible moral approaches like expressivism or error theory than moral realism, but the statement all ethics is simply evolutionary biology seems very premature.”

This selection is probably the most interesting interpretation/rebuttal of ‘reductionism’, a concept commonly used but rarely well-defined. It claims that the formal sciences are in some way ‘autonomous’; they give themselves their own principles. Frege famously defended this in his conflict with Husserl’s early psychologism. According to this modern variety of “Platonism”, the principles of a formal science cannot be derived from any empirical field. If this is true of mathematics and logic, then is seems as though this sets a precedent for the “is/ought” distinction.If a priori knowldege were shown to have a separate basis from a posteriori knowledge, then it could be used to clarify the separation of fact and value. It would seem that Platonism in this sense give us at least two realms of beings who are independent and yet have a certain level of ‘pre-established harmony’ between them. Math for example, is useful and authoritative for many empirical fields from physics to economics. How this could be so was what Kant sought to explain, and his solution sought to bring both formal and ethical beings into relation with the empirical.

My basic idea for naturalistic metaphysics is this: mathematics performs a cognitive function and therefore has adaptive value in light of this function. In order to perform this function, it needs to satisfy certain formal conditions. We are constrained by the definitions of mathematical beings because changing those definitions in the least destroys the functional and adaptive value of math.

Arithmetic is founded on nothing but **the set of sets whose members map onto each other**. “Mapping” means that each member of one set has a unique counterpart in every other set with the same number of members. This is the only way to clearly and primitively define integers, and all other math is founded on this simple set of interrelated definitions. If you change the defintion of one interger, it becomes the same as its neighbor and leaves a gap. Thus there is only one possible set of integers, and therfore only one possible way to relate them, meaning there is only one possible multiplication table.

If you changed the definition of one of the integers, it would also lose it’s adaptive value. Mathematics is only “autonomous” because each mathematical being has NO autonomy from most if not all mathematical beings, and all are dependent on the simple idea of the relationship of sets mapping onto each other. This idea is in turn is derived from problems like figuring out:

- how to share a big basket of fruit
- if a war party is evenly matched by the enemy
- if someone has stolen some of our cattle

If basic arithmetic can’t accomplish this, it’s useless crap. I’m sure I oversimplified and left out some other items ( such as the fact that there are multiple mathematical foundational theories ), but I hope this clarifies my overly-short answer of how evolution can produce creatures with *a priori* knowledge of the ‘universal language’ of mathematics. In this view, some ethical principles can have something like this sort of a priori validity if we can find a suitable starting point. For math, the starting point I used was defining the integers through set theory. The starting point for ethics could start with the game theory of the iterated Prisoner’s Dilemma. In this dilemma, there are only a finite number of Evolutionary Stable Strategy-schemas, i.e., definable sets of strategies that work. Examples of a strategy-schemas include “Initially Benevolent Strategies” ( strategies that do not betray without being betrayed first) and ” 2-turn Forgiving Strategies” ( strategies that stop betraying in revenge after two turns without a betrayal ). Perhaps there’s a better starting point than the Prisoner’s Dilemma. But this is just to illustrate possibility of how to get started generating ethical rules from evolutionary game theory. This could fulfill the dream of Plato, Kant and other ethical rationalists while paying proper respect to modern science.

This is a discussion I had about this post on FB:

James Harley “To try to reduce mathematics to simple arithmetic is laughable.

Appreciate the attempt to confront the devastating argument against naturalistic morality.

Part of Mathematics is the ability to theorize and reason the Limits of what arithmetic machinery can do, the classic example here is Gödel’s incompleteness theorem. There is a difference between understanding Mathmatics and the mindless operation of Mathmatics ( like a computer ).

Adam Voight “Let’s say that I can’t reduce math to arithmetic; what then?

How does incompleteness have anything at all to do with naturalism? Has anyone ever written anything anywhere which explains this?

Are you saying that you have evidence that physics / evolution unaided cannot produce mathematicians? Why are you saying this? Is this one of your gut feelings? Are you simply going to “refuse to believe something”? I’m just going with the assumption that we evolved from apes, and that our math evolved from whatever counting goes on in earlier primates. If math cannot be reduced to arithmetic, that’s fine with me. After all, morality can’t be reduced to care or Utilitarianism. Even more, value cannot be reduced to ethics. All this means is that ethics has more than one cognitive module. There is, in fact one moral module that is distinctively religious- “purity”. But is has a completely naturalistic origin. Perhaps math has more than one cognitive module. Does that refute naturalism? No.

The only thing that could refute naturalism is something nonnatural. What’s non-natural about someone doing math? Is the actual process of doing math supernatural? Does it break the laws of physics? Or is it simply beyond the reach of naturalistic evolution? Is math’s location on the adaptive landscape on the far side of a unbridgeable “trough”?

What is so unnatural about mathematics? People whose “purity” module are highly religious, but they are merely data for me, not an authority. So also with Godel himself; so what if he assumes that numbers popped out of the MInd of God? Can he make an neurologist understand what he’s talking about? If the brain can physically compute Godel’s theorem, and the process of evolution can physically produce mathematical creatures, that is all I need, regardless of whether maths can be reduced to one kind of math.

What is so unnatural about mathematics?

James Harley “1. Glad you agree mathematics is not reducible to simple arithmetic .2 To say mathematics is the result of evolution is not science. It is not falsifiable proposition and credits the label ‘evolution’ as some force in the universe which is not even science. This proposition made the ability to do mathematics is from evolution looks like a statement made by religious fundamentalists which instead of God believes in “evolution”. I should add that I do think there we have some minimal understanding of what is going on by what the label evolution is pointing to, like horizontal gene transfer etc, but we only scratching the surface here. 3. I don’t have an explanation for how we can reason. I think this is most honest answer instead of believing in fairy tales like evolution. I take reason as part of the natural world.”

Adam Voight “It is falsifiable. Just prove that it is neither an adaptation nor a by-product of an adaptation. You do that using the normal methods of biology. If you can’t prove that, then we are warranted in believing that everything about every living creature is either an adaptation nor a by-product of an adaptation. If you could prove this about ANY living creature, it would shake the world. Math and ethics are not really mysterious at all. What about getting depressed and killing yourself? THAT is a real mystery. So why not say that suicide refutes naturalism? It be more persuasive, because you would be claiming to have proven that science will never explain it.

Every scientific theory deserves this same level of respect in its own field. Since we are talking about the adaptive behaviors of living creatures, we are square in the middle of evolution’s proper domain.

James Harley “The various evolutionary theories in “evolution” like random genetic mutation, gene drift, horizontal gene transfer, … are claims about mechanism by which evolution may have occurred. To simply state that a feature is adapted to the environment without any mention which evolutionary mechanism you are referring does not ‘prove’ evolution at all. At present we have no idea how consciousness or reason came to be in biology which I agree is well adapted to the environment.”

Adam Voight “Part of saying something nontrivial and empirically meaningful is to make predictions about what sorts of things must be the case in order for my theory to be true. All I’m saying is that there must be some physical process P that completely implements cognitive process C such that C is identical with any possible mathematical discovery, any ethical judgement/action, any artistic experience, and any religious whatever.That’s a very comprehensive and meaningful statement. The denial of this is that there will be forms of consciousness that have no material correlation whatsoever; kind of like if there were no complex molecules within a cell. In that case, we would have no material basis for life. Our only fall back would be biological non-naturalism a.k.a..”vitalism”. The same could easily be true of any other field in which non-naturalism seems tempting.

On the other hand, what’s the non-naturalist equivalent? Can non-naturalism per se say anything like this? “Something like X must be the case, or I’ll eat my hat.” No, Even if robots end up just like humans in every way, including writing mystical poetry, it won’t refute non-naturalism because there’s nothing to refute, unless you’re claiming that science will grind to a halt now that we have reached the limits of naturalism. If all the non-naturalists in the world would just sign a statement to that effect, then i would respect them somewhat, since they would actually by saying something and sticking to it. All they would have to do is argue their case with enough vigor such that future non-naturalists would be convinced. They would be taking responsibility for some outcome that could possibly be known with certainty. Just doing that would be cause for congratulations.”

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