People overlap.

People overlap.

Many of you will not understand me when I say this. Perhaps it will conjure up images of kissing or having sex, but I refer to the fact that the minds of people overlap.

Brains, of course, do not overlap, so it means something different that minds overlap. That we can think the same thoughts and perceive the same objects is common sense, but for some reason perceptions are regarded as separate; ideas are regarded as separate. And therefore, minds are regarded as separate, and not overlapping.

The notion that minds do not overlap is reinforced by the objective view; it equates minds with brains. Minds as subjectively experienced, however, are the world itself. We experience the world; the neurons that make up our mind are subjectively significant because they represent the world, not because they exist within the space of our bodies. So in a very real sense, by identifying with our minds in addition to our bodies, we acknowledge that we literally extend beyond our bodies; and it is these minds that overlap with one another.

The fact of human overlap cannot be denied, although it can be obscured by a purely materialistic understanding. So for this reason, may we develop our understanding of mind, especially in times when everything appears fragmented and people behave selfishly.

A Common Framework for Cognitive Science

This article was published on the Cognitive Science Society blog.

Cognitive science reports on many mental issues, but there is no model of the mind on which all cognitive scientists agree: at least explicitly.

In this essay (published at https://cognitivesciencesociety.org/framework-cognitive-science/), I argue that cognitive science does in fact have a basic model which goes beyond either behaviorism or even the theory of categories which has been formalized into categories. This basic model is described throughout the book, so this brief essay explains the psychological context from which the model can be understood.

Theories of Cognitive Categorization

Within Cognitive Psychology, there is much debate about theories of categorization. Models such as Prototype Theory (popularized by Eleanor Rosch) and Exemplar Theory (popularized by Robert Nosofsky) are probably the most popular, and are both different from Aristotelian Categories.

With respect to the basic model of cognition, categories are defined both in terms of concrete unions and abstract intersections. So, the category of fruits may be defined as either a concrete category (in which case it is a combination of experiences of individual fruits), an abstract category (in which case it consists of those properties common to the categories which participate in its definition), or some combination of both.

In Professor Rosch’s later work, she felt that many experimental paradigms were not sufficient to distinguish between prototype and exemplar theory. Similarly, I am not sure how a theory that incorporates both aspects of categorization can be effectively tested.

If anyone has a proposal for how a theory such as the basic model of cognition proposed in the whole part can be tested, please add to the comments below; I am not very familiar with the associated experimental paradigms.

Infinite Tables

The use of completed infinity can become even more problematic when comparing multiple infinite sets, a practice pioneered by Georg Cantor.  Here we analyze one of Cantor’s arguments commonly called the diagonalization argument.  

According to the diagonalization argument, if T is any infinite sequence of binary digits, then it is possible to use the entries of T to construct an element of T that is not found within T.  Cantor uses this result to prove that the set T is not countable.  His proof begins with an enumeration of elements from T; here, we use the series generated by counting in binary from zero to one, where there is an implicit decimal point on the left hand side of the entries and the left-most index varies fastest (i.e., [0, 1/2, 1/4, 3/4, 1/8, …]): 

Eq 1:

The binary complement of the nth digit from each series sn is used to create a new series (s), that differs from each element of T by at least one digit: in this example, s turns out to be the infinite-length series of all ones.  

The problem is fairly clear: the number which Cantor proves cannot be found in T, sn, is in fact the very last entry in the completed table. If this proves anything, it proves that the table is incomplete. However, Cantor’s proof requires that a completed infinite number of elements is necessary: if an incomplete table is used, it is not possible to prove that s is not a part of T. This problem can be expressed slightly more formally as follows:

  1. Cantor’s argument relies on a table, T, with a completed infinite number of rows (otherwise he could not show that the generated number was different from all of the rows).
  2. Applying the method of diagonalization to the table used above generates a number sn that is all ones.
  3. sn is the last entry in the completed table.
  4. Therefore, the process of diagonalization did not reach the end of the table.
  5. Therefore, Cantor’s argument is not effective

In other words, Cantors argument produces either a number within the table (the last entry), or it does not reach the end of the table (which is required by the construction).  In neither case does it produce a number different from all entries in that table.

For a finitist, the table size is not allowed to be infinite, which removes the problem (and prevents Cantor’s argument from working). To understand this, note that applying diagonalization to the following completed table of two digits produces a number which is in the table:

Eq 2:

A different way of understanding this argument relies on the fact that diagonalization is only suitable as an argument if the table construction proceeds downwards as fast as it proceeds to the right. This occurs only in tables that use non-position-based number systems. For example, the table which has a unit entry in only one position grows downwards as fast as it grows to the right, and cannot be used in Cantor’s “proof”:

Eq 3:

What exactly does this prove? I leave that up to mathematicians to decide. Personally, I take it to be an endorsement of finitist position in mathematical philosophy, or at least a disproof of Cantors argument. More generally, the problem is not the use of infinity, but rather the use of infinity outside of the context of a limit.  In other words, the problem is treating infinity like a number instead of a process.

Infinite Representation

Personally, I believe in a constructivist approach to mathematics. As such, I do not believe in the notion of completed (actual) infinity: I am a classical finitist, which means I use infinity, but only when it is used in conjunction with a limit. One way in which the use of actual infinity seems problematic to me is related to decimal representation.  In particular, there is a paradoxical statement that is held to be true by infinitists (mathematicians who believe in completed infinity) that numbers can have multiple decimal representations. In their defense, they make the following claim:

Eq 1: 

Most people are not intuitively comfortable with this statement, because it implies that two different numerical (decimal) representations correspond to a single number (and I think that intuition is correct).  The “proof” of identity begins by assigning an infinitely repeating value to a variable x as follows:

Eq 2:

Multiply both sides by 10 to get: 

Eq 3:

Subtract (2) from (3) to get: 

Eq 4:

Equating x in (2) and (4) leads directly to (1).  

Infinitists take the position that this is not paradoxical; they simply accept that a single number has multiple decimal representations, regardless of whether that result is intuitively paradoxical.  

Classical finitists can avoid the paradox by handling infinite representations as limit series.  As a result, believers in potential infinity are not committed to a single number having multiple equivalent decimal representations.

The essential difference in approach is that believers in potential infinity refuse to represent a number which has actual infinite length, so a representation where the nines run “to infinity” is not acceptable.  The potentially infinite representation is therefore written as a limit series:

Eq 5:

In other words, repeating decimal representations are written as a limit where the number of terms approaches infinity.  The significant difference between the limit representation and the repeating decimal representation is that the former is clearly a process, not a number whose length has a cardinality of aleph-null.  

Using the limit representation in (5), equation (4) may be re-written as:

Eq 6:

Note that (6) does not reproduce the result of (4) (i.e., that x=1). 

The different results obtained by using the limit formula as opposed to the overbar formula are the result of being forced to take account of the number of terms in the series expansion.  In other words, we did not take account of the term in the subtraction in (4), because it was assumed that the decimal string was of infinite length.  Therefore, the term i in the second series expansion is subtracted from term i+1 in the first series expansion.