March 29, 2012
Morality and Mathematics: Can You Be A Moral Antirealist and a Mathematical Realist?
Justin Clarke-Doane in Ethics (via the NYT):
It is commonly suggested that evolutionary considerations generate an epistemological challenge for moral realism. At first approximation, the challenge for the moral realist is to explain our having many true moral beliefs, given that those beliefs are the products of evolutionary forces that would be indifferent to the moral truth. An important question surrounding this challenge is the extent to which it generalizes. In particular, it is of interest whether the Evolutionary Challenge for moral realism is equally a challenge for mathematical realism. It is widely thought not to be. For example, Richard Joyce, one of the most prominent advocates of the Evolutionary Challenge, goes so far as to write, “the dialectic within which I am working here assumes that if an argument that moral beliefs are unjustified or false would by the same logic show that believing that 1 + 1 = 2 is unjustified or false, this would count as a reductio ad absurdum.”1 He assures the reader, “There is … evidence that the distinct genealogy of [mathematical] beliefs can be pushed right back into evolutionary history. Would the fact that we have such a genealogical explanation of … ‘1 + 1 = 2’ serve to demonstrate that we are unjustified in holding it? Surely not, for we have no grasp of how this belief might have enhanced reproductive fitness independent of assuming its truth.”2 Similarly, Walter Sinnott-Armstrong writes, “The evolutionary explanations [of our having the moral beliefs that we have] work even if there are no moral facts at all. The same point could not be made about mathematical beliefs. People evolved to believe that 2 + 3 = 5, because they would not have survived if they had believed that 2 + 3 = 4, but the reason why they would not have survived then is that it is true that 2 + 3 = 5.”3 Finally, Roger Crisp writes, “In the case of mathematics, what is central is the contrast between practices or beliefs which develop because that is the way things are, and those that do not. The calculating rules developed as they did because [they] reflect mathematical truth. The functions of … morality, however, are to be understood in terms of well-being, and there seems no reason to think that had human nature involved, say, different motivations then different practices would not have emerged.”4
In this article, I argue that such sentiments are mistaken. I argue that the Evolutionary Challenge for moral realism is equally a challenge for mathematical realism.
Posted by Robin Varghese at 09:29 AM | Permalink
Comments
This seems interesting. Can anyone translate a summary into English for me?
Posted by: Eli | Mar 29, 2012 11:31:05 AM
Isn't the question can you be real, but still dabble in "reason" a bit?
Posted by: Dredd | Mar 29, 2012 2:23:56 PM
Eli,
Don't feel bad. I don't understand it either..
I see that Prof. Sokal's practical joke some years back did not put an end to this type of impenetrable blather.
Posted by: Bill | Mar 29, 2012 4:45:24 PM
Hmm I believe that "mathematical realism" isn't as secure as one might think. For instance, there are different approaches to the foundations of mathematics that disagree on certain principles.
During the 1920s there was some debate over constructivism, the notion that one had to construct a mathematical object to prove it exists. Constructivists reject the law of the excluded middle (that a given mathematical statement is either true or false), so proving that the non-existence of an object leads to a contradiction is not enough, in the view of constructivists, to prove that the object exists. Famously, the mathematician David Hilbert forcefully rejected constructivism, insisting on the usefulness of the excluded middle. The debate was never strictly speaking resolved, but has sort of settled down, with constructivist proof treated an object of study in proof theory.
Associated with constructivism is intuitionism, the notion that mathematics ultimately rests not on some absolute truth but on the intuitions of the mathematician. For instance, finitists dispute the "existence" of infinity. There are even ultra-finitists, who are even suspicious of large numbers. A story about Soviet dissident mathematician Alexander Yessenin-Volpin:
The history of set theory is also revealing. The original set theory of Frege, Russell et alii simply defined a set as a predicate. If the predicate applied to an object, it was a member of the set, otherwise it wasn't. Simple enough, but then Russell noticed that the predicate "is not a member of itself" led to a contradiction. A great deal of mathematical effort was then spent coming up with something consistent, and the original effort is now referred to as "naive set theory".
These days the most used set theory is an axiom-based system called Zermelo-Frankel. But there's an additional axiom known as the "axiom of choice" that was at one time controversial. Basically, if one has a collection of non-empty sets, there exists a collection of elements one from each set, even when it's an infinite collection and you can't construct a method of choosing elements.
It's due to the axiom of choice that, for instance, there "exists" a way of splitting a ball into five pieces and reassembling them into two balls of the same size, even though one cannot construct the pieces mathematically. This isn't possible without AC, because then every set has a measure (in three dimensions, volume).
Posted by: Sagredo | Mar 30, 2012 5:02:37 AM
Yep, this essay is frustrating in all the ways that analytic philosophy can be frustrating: The narrow focus on anlyzing language, the absence of context, the failure to really engage with other disciplines despite ostensibly addressing "the evolutionary challenge".
It seems to boil down to "some people who are skeptical about apples assume the existence of oranges. But I think if you you are an apple anti-realist, you should be an orange anti-realist too."
But why? Moral statements endorse reasons for action. Mathematical statements identify relationships between ratios. We indicate something very different when we describe a moral statement as "true" than when we describe a mathematical statement as "true".
In that case, why can't you be a moral anti-realist and a mathematical realist? And mightn't our evolutionary history affect our actions and our beliefs about what counts as a reason for action in a way that it doesn't affect either mathematical relationships or our knowledge of those relationships?
An example: A commonly used example of a moral axiom that involves the value of children. E.g. it is "wrong to murder innocent children."
This is certainly a widely held and deeply felt intuition. A moment's reflection reveals that young children are neither more sentient nor more rational than animals we routinely butcher and eat. So where does it come from?
It has an obvious evolutionary cause: Mammals are born with undeveloped brains so grown-ups have a strong instinct to protect them until they come online. (Birds are like this too, because it takes time for them to learn to fly.)
But insects, fish, amphibians and (most) reptiles don't have (or need) this instict.
A flight of fancy: Is it absurd to imagine a species of spiders involving human level intelligence? If they did would they develop a taboo against, e.g. eating your spouse / parents / offspring? Or would they keep behaving like real, simpler spiders do? (Think about what basic instincts we share with other mammals and other primates vs. how we differ from them.)
If they retained (and, being intelligent, endorsed) certain spiderly behaviors, what would we make of their "morality"?
We might say that their moral intuitions were "true" for them. In that case we accept a rather extreme kind relativism.
Perhaps we would instead say that they held moral beliefs that were systematically false. But all we would mean by that is that they hold beliefs different from our own.
My own personal view is that moral statements don't really have a truth value in the sense of corresponding to any brute fact. To say "It's true that, e.g., you should protect children" can only mean (depending on the context) "I endorse protecting children", or "People around here follow a rule of protecting children", or "I count some state of affairs, (e.g. pleasure, human happiness) as a reason for action, and believe that protecting children promotes that state of affairs." Facts are involved, but not "moral facts". There are no moral facts.
Posted by: JoshM | Mar 30, 2012 9:05:06 AM
"Facts are involved, but not "moral facts". There are no moral facts."
And you're ok with the fact that you offer no argument whatsoever for this?
Posted by: MRM | Mar 30, 2012 9:45:54 AM
It would help to understand what morality is before writing about it. Morality is those actions and behaviors that lead to the good health and well-being of individuals and communities. Thus, morality is just as much a part of natural law as is mathematics.
Mathematics were not invented by human minds. Mathematics is an inherent part of the structure underlying the whole Universe. Humans discovered this realization and developed expressions for it.
The same goes for morality. Life requires certain conditions to flourish. Those conditions that lead to good health and happiness encourage a life form to perpetuate its species. Religions are like mathematics in that they were developed by humans to codify an expression of the healthful and well-being behaviors and thus manifested morality.
Morality is an essential and central purpose of government. Every law, rule, and regulation is a legislation of morality. Each is intended to insure the health and well-being of individuals and communities.
Posted by: David Thomson | Mar 30, 2012 9:45:56 AM
What soulless blather. The reason 2 exists isn't that we all agree it does. It just does. Pretending moral truth is the same? That's just evil.
Posted by: Jameson Quinn | Mar 30, 2012 1:21:06 PM
The intuitionists would disagree with this. The only mathematics we know comes from human minds. Unlike science, it doesn't rely on sensory observation. And there is some disagreement over mathematical truth.
Posted by: Sagredo | Mar 30, 2012 9:39:38 PM
How do you know?
Posted by: Sagredo | Mar 30, 2012 9:41:06 PM
Mathematics is just a language and a modeling tool invented by humans to describe the quantitative aspects about the world just a as say the English language is used to describe it qualitatively. Planets orbit and birds fly without knowing about mathematics. Yes there is logic and consistency in how nature behaves (or so it seems for most of the time) and this can be discovered through dialectics as was done for ages. Mathematics is just a more efficient language for the mind as say Assembly language is compared to COBOL for computers.
Posted by: Raza | Mar 30, 2012 11:04:14 PM
Post a comment