On Counting (Exploring Operational Definitions Part III)

For the first two parts in this series see:

Exploring Operational Definitions: Part I
Exploring Operational Definitions: Part II - Distance

Perhaps the "simplest" procedure that most folks have learned is the technique(s) of counting. What I would like to explore is that there are a variety of techniques that we call counting. In some cases they build on one another. In other cases, they are techniques which give "approximate" solutions.

Of course not all societies count things (see here). Nonetheless, I suspect that many of our "intuitions" about mathematics ultimately stem from our earlier experience with counting. Our attachment to such intuitions will somewhat determine how willing we are able to accept alternative definitions and techniques for counting. Today I'll explore a few of these definitions.

Exploring Operational Definitions: Part II - Distance

So in Part I, I began discussing operational definitions. In particular, I've been focusing on the concept of distance.

I proposed that we might consider our intuitions regarding answering the question - what is distance? - by means of early experience of operational definitions of distance. The idea being that what we consider distance is going to be closely related to ways we were taught to measure distance.

Correlation as a Substitute for Critical Thinking in Finance?

I frequently come across graphs like these on various blogs and articles. I'm going to single out one, not because there is anything particularly wrong about it (they're all wrong), but only to use it for illustrative purposes. I could really pick out any of these and offer the same criticisms.

This one I found in the article The Declining Inflation Expectations Chart That Should Have Stock Investors Very Concerned. The original chart apparently come from Dan Greenhaus via twitter.

Exploring Operational Definitions: Part I

An operational definition is a way to define a concept by the set of operations or procedures which are used to make a judgement with that concept. This idea, along with a philosophical thesis on meaning known as operationalism, was popularized by Percey Bridgman (see his The Logic of Modern Physics).

One example he liked to use is the concept of distance.

On [Logical] Equivalence

In Newton's Law is F=ma?, explored the issue of whether or not this formulation was equivalent to Newton's actual statement of his 2nd Law. Today I want to further explore the topic of [logical?] equivalence. In general, I want to know what it means to say that two statements, $P$ and $Q$, are equivalent.

Newton's Law is F=ma?

This is a brief exploration on the logical structure of Newtonian mechanics. 

When physicists teach classical mechanics, they often refer to Newton's 2nd Law as "F=ma" or more technically as:

The Structure of Arguments: Deduction, Induction and Abduction

So this is a brief overview of three types of logical arguments.

Plantinga on Antirealism

So I decided to read Alvin Plantinga's How to be an Anti-Realist which I thought might be interesting. Unfortunately I was disappointed.

I won't present a thorough discussion of the paper but I will draw upon one bit of reasoning I found a bit goofy.

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.

On the Subject Matter of Philosophy

Most disciplines have a very specific, known (and in some cases quite narrow) subject matter. The subject matter of the discipline specifies a domain over which research might take place. For biology, for example, the Greek origins of the term suggest the domain of study is life. Psychology is the study of the mind.

While it's not always the case we can define clear-cut boundaries, there are definite subject matters which we can point and say "that's a question that the psychologists should investigate".

This raises the question of what counts as the genuine subject matter of philosophy. In a broad sense, the term philosophy means "love of wisdom". It seems to encompass all that might be learned or discovered. For example, in its early stages, physics was often referred to as "natural philosophy".

While I am very much sympathetic to this idea, in practice, philosophers have tried to separate their discipline from other disciplines. So what is it that makes philosophy a unique discipline?

Gettier Intuition

One common criterion used to determine whether or not something is knowledge is the "justified true belief" criterion (JTB). It has many forms but the basic idea is this:

S knows that P means:

1. S believes that P.
2. S has justification for believing P.
3. S is true.

In 1963 Edmund Gettier proposed counterexamples to the above relationship. The idea is pretty simple. Construct a scenario in which (1), (2) and (3) are satisfied, but it fails to be knowledge. We'll look at an example from his original paper: Is Justified True Belief Knowledge? In particular, we'll look at Case I.

That's an Empirical Question: Part IV

This is the 4th part in a series called "That's an Empirical Question". For the other parts of this series see here:

Part I
Part II
Part III

I think this one deserves an appropriate subtitle.

Is Induction Always Empirical?

In Part II, I suggested that induction is an important "empirical tool". Assuming that's true, that still raises the question whether all uses of the "induction tool" necessarily amounts to an "empirical investigation".

To explore this, as per usual, I will use an example: The Goldbach Conjecture.

That's an Empirical Question: Part III

This is a follow-up in a series exploring what the difference between "empirical" versus "mathematical" versus "philosophical" questions. For the other posts in the series see:

That's an Empirical Question: Part I
That's an Empirical Question: Part II

I'd like to consider a thought experiment though I think you can probably find some similar at manufacturing firms in the real world.

Do philosophers know particulars?

In Methodists Vs Particularists, I suggested that philosophers who are particularists may not, in fact, be so. It's possible that there are no particulars that are known per se but rather a broad set of "methods" that are employed. This may not be explicit criterion per se; as a result I shall refer to these simply as "heuristics".

We'll consider a common "simple" example in philosophy: color identification. I believe that most children learn color identification via ostensive definitions. Consider the following scenario:

Methodists Vs. Particularists

In Intuition in Philosophy, I suggested that one difference between, say, mathematicians and philosophers is that philosophers will willingly privilege an intuition over a derived result whereas the mathematician will reject the intuition in favor of the derived result. To some extent I think this characterization may not have been entirely accurate. At the very least some clarification needs to be made.

I here will argue that many philosophers are particularists whereas mathematicians tend to be methodists. "Intuition" plays a role in what particulars the philosopher claims to have knowledge. This allows for there to be something along the line of "mathematical intuition" of which I think is believed to be present by many mathematicians. I will briefly sketch out the issue here.

Was Chess Invented or Discovered?

One common comparison made within philosophy of math circles is a comparison between the game of chess and mathematics. Variants of, in particular, formalism hold that mathematics is a sort of game with well defined rules that mathematicians play. I think the game of chess is actually quite apt for the comparison. Like all metaphors, it has limitations but I think it's still an interesting one.

The closest comparison, in my view, is between chess and geometry. Here's how it works.

Being Without Numbers

Linguistic anthropologist Daniel Everett has been researching and living with (off and on) an Amazonian tribe called the Pirahã for a few decades now. In Don't Sleep, There Are Snakes: Life and Language in the Amazonian Jungle, Everett gives a semi-autobiographical account of his interactions with the Pirahã.

Intuition in Philosophy

Intuition has an interesting role in philosophy, particularly analytic philosophy. One of its functions is epistemic. It's used to test various assumptions and conclusions in philosophy for validity or truth. If something fails to stand up to intuition the propositions in question get dismissed.

There's also a good deal of literature in a field known as "experimental philosophy". The research often takes the form of surveys to different groups of individuals (intro to philosophy students, professional philosophers, etc). The purpose of the surveys is to assess what "intuitions" people have. What the research tends to indicate is that intuitions vary greatly between groups depending upon a variety of circumstances (culture, philosophical exposure, etc.)

Modeling Gold Returns: A Case Study

In my previous post, I considered the problem of induction. Looking at emeralds and noting that they are green (or grue) is supposed to be a simple example since making color judgements is presumably a simple task. The example could be further complicated if we factor in the vagueness of making color judgements. After all, it's likely the case that the distribution of electromagnetic frequencies differ between one emerald and the next. (And they would differ further depending on the distribution of EM frequencies of the "white light" used upon it.)

In most real world scientific inquiries, these uncertainties are present and need to be dealt with. So instead of focusing on the sorts of examples used in "simple" philosophical thought experiments, I thought I would provide a more detailed example.

The motivation for looking at this came from an interesting paper entitled The Golden Dilemma. I will make frequent reference to Exhibits from this paper. [1]

The Problem of Induction

This is a continuation of a series devoted to the question of what an "empirical question" is. See Part I and Part II if you're interested.

One tool that is considered essential to the empirical sciences is induction. It's importance is, in my opinion, overstated by most philosophers (I firmly believe that abduction rarely gets the proper credit it deserves). In spite of that, I do not dismiss its important role in scientific inquiry.