Notes on viewing

## Shared mental objects

I propose the phrase "shared mental object" to name the sort of thing that includes mathematical objects, abstract objects, fictional objects and other concepts with the following properties: ​

• They are not physical objects
• We think of them as objects
• We share them with other people

It is the name "shared mental object" that is a new idea; the concept has been around in philosophy and math ed for awhile and has been called various things, especially "abstract object", which is the name I have used in abstractmath.

I will go into detail concerning some examples in order to make the concept clear.  If you examine this concept deeply you discover many fine points, nested ideas and circles of examples that go back on themselves.  I will not get very far into these fine points here, but I have written about some of them posts and in abmath (see references).  I am working on a post about some of the fine points and will publish it if I can control its tendency to expand into infinite proliferation and recursion.

## Examples

### Messages

There is a story about the early days of telegraphy:  A man comes into the newly-opened telegraph station and asks to send a telegram to his son who is working in another city. He writes out the message and gives it to the operator with his payment.  The operator puts the message on a spike and clicks the key in front of him for a while, then says, "I have sent your message.  Thanks for shopping at Postal Telegraph".  The man looks astonished and points at the message and says, "But it is still here!"

A message is a shared mental object.

• It may be represented by a physical object, such as a piece of paper with writing on it, and people commonly refer to the paper as the message.
• It may be a verbal message from you, perhaps delivered by another person to a third person by speech.
• The delivery process may introduce errors (so can sending a telegraph).  So the thoughts in the three brains (the sender, the deliverer and the recipient) can differ from each other, but they can still talk about "the message" as if it were one object.

Other examples that are similar in nature to messages are schedules and the month of September (see Math Objects in abmath, where they are called abstract objects.).  In English-speaking communities, September is a cultural default: you are expected to know what it is. You can know that September is a month and that right this minute it is not September (unless it is September). You may think that September has 31 days and most people would say you are wrong, but they would agree that you and they are talking about the same month.

The general concept of the month of September and facts concerning it have been in shared existence in English-speaking cultural groups for (maybe) a thousand years.  In contrast, a message is usually shared by only two or three people and it has a short life; a few years from now, it may be that none of the people involved with the message remember what it said or even that it existed.

### Symbols

symbol, such as the letter "a" or the integral sign "$\int$", is a shared mental object.  Like the month of September, but unlike messages, letters are shared by large cultural entities, every language community that uses the Latin alphabet (and more) in the case of "a", and math and tech people in the case of "$\int$".

The letter "a" is represented physically on paper, a blackboard or a screen, among other things.  If you are literate in English and recognize an occurrence as representing the letter, you probably do this using a process in the brain that is automatic and that operates outside your awareness

Literate readers of English also generally agree that a string of letters either does or does not represent the word "default" but there are borderline cases (as in those little boxes where you have to prove you are not a robot) where they may disagree or admit that they don't know.  Even so, the letter "a" and the word "default" are shared in the minds of many people and there is general (but not absolutely universal) agreement on when you are seeing representations of them.

### Fictional objects

Fictional objects such as Sherlock Holmes and unicorns are shared mental objects.  I wrote briefly about them in Mathematical objects and will not go into them here.

### Mathematical objects

The integer $111$, the integral $\int_0^1 x^2\,dx$ and the set of all real numbers are all mathematical objects.   They are all shared mental objects.  In most of the world, people with a little education will know that $111$ is a number and what it means to have $111$ beans in a jar (for example).  They know that it is one more that $110$ and a lot more than $42$.

Mathematicians, scientists and STEM students will know something about what  $\int_0^1 x^2\,dx$ means and they will probably know how to calculate it.  Most  of them may be able to do it in their head.  I have taught calculus so many times that I know it "by heart", which means that it is associated in my brain with the number $1/3$ in such a way that when I see the integral the number automatically and without effort pops us (in the same way that I know September has 30 days).

Beginning calculus students may have a confused and incorrect understanding of the set of all real numbers in several ways, but practicing mathematicians (and many others) know that it is an uncountably infinite dense set and they think of it as an object.  A student very likely does not think of it as an object, but as a sprawling unimaginable space that you cannot possibly regard as a thing. Students may picture a real number as having another real number sitting right beside it — the next biggest one. Most practicing mathematicians think of the set of real numbers as a completed infinity – every real number is already there —  and they know that between any two of them there is another one.

As a consequence, when students and professors talk about real numbers the student finds that some times the professor says things that sound completely wrong and the professor hears the student say things that are bizarre and confused.  They firmly believe they are talking about the same thing, the real numbers, but the student is seen by the professor as wrong and the professor is seen by the students as talking meaningless nonsense.  Even so, they believe they are talking about the same thing.

## Nomenclature

I tried various other names before I came to "shared mental objects".

• I called them abstract objects in abstractmath.  The word "abstract" does not convey their actual character — they are mental and they are shared.
• They are non-physical objects, a phrase widely used in philosophy, but naming something by a negation is always a bad idea.
• Co-mental objects is ugly and comental looks like a misspelling.
• Intermental objects sounds like it has something to do with burial.  Maybe InterMental?
• The word entity may avoid some confusion caused by the word "object", which suggests physical object.  But "object" is widely used in philosophy and in math ed in the way it is used here.
• Meme?  Well, in some sense a shared mental object is a meme.  Memes have a connotation of forcing themselves into your brain that I don't want, but I want to consider the relationship further.

The major advantage of "shared mental object" is that it describes the important properties of the concept: It is a mental object and it is shared by people.  It has no philosophical implications concerning platonism, either. Mathematical objects do have special properties of verifiability that general shared mental objects do not, but my terminology does not suggest any existence of absolute truth or of an Ideal existing in another world.  I don't believe in such things, but some people do and I want to point out that "shared mental object" does not rule such things out — it merely gives a direct evidence-based description of a phenomenon that actually exists in the real world.

## References

Abstract objects in the Stanford Encyclopedia of Philosophy

Abstract object in Wikipedia

Mathematical objects in abstractmath

Mathematical objects in Wikipedia

What is Mathematics, Really?  R. Hersh, Oxford University Press, 1997

Previous posts

Representations of mathematical objects

Representations III: Rigor and Rigor Mortis

Representations II: Dry Bones

### Notes on Viewing

This post uses MathJax. If you see mathematical expressions with dollar signs around them, or badly formatted formulas, try refreshing the screen. Sometimes you have to do it two or three times.

1. 1
eddie says:

I can't see any difference between an "abstract object" and a "mathematical object" except for their qualities, Sherlock Holmes and Unicorns have their unique qualities, just as numbers do.

• 2
SixWingedSeraph says:

I wrote about that here: http://www.abstractmath.org/MM/MMMathObj.htm#_Toc153180903   Mathematical objects are abstract objects, but abstract objects can change over time, among other things.  Math objects are unchanging and don't affect anything.  That distinction is not mine —  it is made by some philosophers — but of course you can argue about whether it is correct or not. –CW