## Sunday, April 7, 2013

### On the Existence of Married Bachelors

In my series on That's an Empirical Question, I noted that philosophers often consider questions which are "empirical" as being outside the scope of philosophy. The domain of philosophy would then be some subset of "non-empirical" questions. Part of my goal has been an attempt to demarcate the empirical from the non-empirical.

Today I will continue this endeavor by exploring it from the other side: what makes a question non-empirical and is there a place for these sorts of questions. This is a very large subject, which could not be treated in one blog alone. I will, however, start with an example (as the title of the blog suggests): All bachelors are unmarried men.

As the title of my blog suggests, I'd like to suggest that it's possible for married bachelors to exist. This flies in the face of the received wisdom that there can be no married bachelors because the following claims are taken as given:
1. Bachelor is defined as "unmarried man".
2. There is no reason to revise a given definition.
While I have no objections to the first claim, I do have objections with the second. I will spell this out with a couple of examples.

Example 1: Momentum

Here I'm referring to the concept in physics known as momentum.

For those of you who are not mathematically inclined, Hang Your Head in Shame, and feel free to proceed to the second example.

Newton referred to this as a "quantity of motion". According to my translation of his Principia (see here), Newton defined the quantity of motion as:
THE QUANTITY OF MOTION IS THE MEASURE OF THE SAME, ARISING FROM THE VELOCITY AND QUANTITY OF MATTER CONJUNCTLY.

The motion of the whole is the sum of the motions of all the parts; and therefore in a body double in quantity, with equal velocity, the motion is double; with twice the velocity, it is quadruple.
Newton had in mind a concept fairly close to the classical notion of momentum:

$$p = m\vec{v}$$

In particular he notes that momentum is a function of mass and velocity. He further notes that mass is linearly related with momentum. What isn't clear from this definition is the last part when he states that double velocity results in quadruple the momentum. This very well is a continuation of the same premise (double the mass) and therefore indicating that he thought velocity was linearly related with momentum. In any event, classical physicists eventually agreed upon the above definition and that's all that matters for the moment.

Newtonian physics can derive an important result: Conservation of Momentum (Corollary III). He stated it rather cumbersomely but we can forgive him for that; he didn't have the tools of vector analysis available to him. For a quick and dirty derivation we need his second and third laws:

Newton's Second Law

$$\vec{F}=\frac{d\vec{p}}{dt}$$

Newton's Third Law

If there are two particles, 1 and 2, which interact, then the force of the interaction is equal and opposite:

$$\vec{F_1} = -\vec{F_2}$$
Combining the two laws we get:
$$\frac{d\vec{p_1}}{dt}=-\frac{d\vec{p_2}}{dt}$$
or
$$\frac{d(\vec{p_1}+\vec{p_2})}{dt}=0$$
And since according to Newton "the [momentum] of the whole is the sum of the [momentum] of all the parts", we have conservation of momentum.

So classically, momentum was an important quantity. And then Einstein came along and changed everything.

In order to maintain conservation of momentum, momentum has to be given by the following equation:

$$\vec{p}=\frac{1}{\sqrt{1-v^2/c^2}}m_0\vec{v}=\gamma m_0\vec{v}$$
where $m_0$ is the rest mass.

Now this either means (at least) one of three things:
1. Momentum ($\vec{p}=m\vec{v}$) is not conserved.
2. Momentum cannot be defined as $\vec{p}=m\vec{v}$.
3. Mass is a function of velocity and the rest mass ($m=\gamma m_0$).
There is a general consensus amongst physicists that (1) should not be adopted. After all, if momentum isn't conserved, what point is there to talk about momentum at all? It would be a useless concept all around.

In addition, since we noted that it could be derived from Newton's 2nd and 3rd Laws, we'd have to abandon or modify those. So (1) is not the best option here.

There is some disagreement, however, over whether to accept (2) or (3). Some prefer (3) or what is known as "relativistic mass". The benefit here would be that you can then sort of keep the classical definition of momentum.

Other physicists reject the notion of "relativistic mass" and therefore have to accept that the classical definition of momentum is no longer applicable.

The point of this example is hopefully obvious by now. How we choose to define things matters and we may later need to alter our definitions to something more useful.

Example 2: Unicorn

As the name suggests, unicorns have only one horn (see here). Unicorns are often represented as horse-like creatures. They sometimes even possess magical powers (such as the ability to not have to wait in line at the grocery store).

We know of unicorns as they occur in various cultural media such as literature and art. Some believe that they may have been loosely based on some (perhaps extinct) real biological creature.

As a result, I would suggest that, while the definition may be a mere matter of convention, there is nothing arbitrary about it. If we are attempting to discover discover some biological creature for which unicorn myth arose out of, that will guide our definition.

But even if we abandon the notion of any biological unicorn, that doesn't mean we can define unicorns as anything we'd like.  Unicorns have an important presence in literature. If I defined a unicorn as "a pig with purple wings, green poca dots and the ability to do calculus", there probably wouldn't be too many unicorns in literature.

We could define it that way, but the word would no longer carry any meaningful usage. It would likely be abandoned altogether. So there are constraints on how we can define unicorns and that will be dictated, say, by unicorns found in literature.

Back to Bachelors

So lets take this back to our bachelors. At present, the statement "all bachelors are married" is entailed by our definition of bachelor. But it could so happen that, for example, our institutions of marriage evolve so much over time that the term could become useless. At that point we could either:
1. Insist that bachelor implies an unmarried man and simply stop using the term or
2. Redefine bachelor (or allow the term to evolve) in response to the changes in our social institutions.
And I'm not sure if it really matters which one we do. It may be worthwhile to keep the term bachelor defined as "unmarried man" so that we can talk about bachelors in their historical context.

Concluding Thoughts

Statements of the form "all bachelors are unmarried men" are no doubt true. But their truth is a matter of convention. We stipulate various definitions and use terms in particular ways and there is something contingent about these definitions. This is not to say such definitions are arbitrary. As the examples above show, there may be valid reasons to prefer one way of defining over another way of defining.

So what does that make philosophy? Is a portion of philosophy devoted to a dialogue over how we define terms? Is it distinct from lexicography? Or is a matter of determining how we ought to define terms? And what's the mechanics of such a dialogue?