Personal Responsibility

I declare:

• the fact of my personal responsibility to acquire or create concepts to understand the world and communicate that understanding.
• the fact of my personal responsibility to create satisfying conditions (whether local or global) for myself and others.

– Note: The word “responsibility” here refers not a duty but rather a “response-ability” and a simple truth: if you want a better world, you must act in ways that are more likely to bring such a world into being. To deny one’s personal responsibility is simply to deny one’s options or the meaningful difference between them.

Personal Independence

I declare:

• the fact of my cognitive and moral independence based upon my own values and judgement.
• the fact of my political independence to set the foundations and context for my own political relationships and to give proper and useful names to behaviors that do or do not conform to my ethics and consent.

– Note: The word “independence” here refers not to an isolation or self-sufficiency but rather a simple truth: your decisions are made in reference to your own values and analysis, to the degree that you are awake, conscious, and thinking. To deny one’s independence is simply to deny one’s own personal values and judgement.

Personal Ethics

I declare:

• my intention to help build and maintain relationships and institutions — be they social, educational, economic, political, legal, or otherwise — that conform to my ethics and philosophy.

Social Contract

I declare:

• my respect for each sane and innocent individual — my intention to treat each sane individual as independent, with the ability, when innocent, to meaningfully and materially withdraw consent from the relationships or institutions that each finds thonself in or under — and my intention to treat each such person as the owner of thon’s own body, being free to choose what to do with one’s own body and life, so long as one doesn’t initiate or apply excessive force, threat, or fraud against others or their (justly acquired) property.
• my intention to conform to the non-aggression principle — that is, to not initiate force, threat, or fraud against others or their (justly acquired) property. (The issues of just acquisition of property and limitations on ownership are worthy of debate and negotiation. The Georgist/geolibertarian position seems to be superior to the roughly Lockean position, if executed well.)
• my intention to reduce unnecessary force, threat, or fraud when possible.

Terminology

I declare:

• myself and every sane and innocent human to be a _sovereign person_, defined to be a person with autonomy and authority over oneself, not being rightfully subject to or owned by any other person or institution, owing allegiance to no one except to those one determines for oneself. A sovereign person may levy defense of thonself and thon’s (justly acquired) property, conclude peace, contract alliances, engage in commerce, freely associate and disassociate with others as one pleases, choose what to do with one’s own body (including what to eat or not, what drugs to take or not, whether to wear clothes or not, and whether to end one’s life or continue or extend it), and so on. Others may apply social pressure in response to these choices, but they may not legitimately respond with aggression (that is, initiation of force, threat, or fraud).

Notes on Consent

From the US Declaration of Independence, we read that “Governments … [derive] their just powers from the consent of the governed…”

Well, consent must be revocable, or able to be withdrawn, in order to be true consent. If one cannot say “no”, then no true consent is present. Our current government thus does not have our consent since we cannot withdraw our material, monetary support without facing the threat of force and violence or its execution. Taxes are extortion and theft, even if most people are happy to submit. (There are better ways to pay for services that have true demand.) Having to leave the country, and leave your loved ones, family, friends, culture, job, etc, only to find another similar government somewhere else, is too high of a bar for measuring consent.

As things stand, most governments are a protection racket — an institution that provides protection with the threat that if you don’t pay for the protection, you will be punished or possibly killed if you resist. This is a moral contradiction and a political failure. We can do better than this.

With a global and a historical perspective, there is much to be grateful for here in the USA, in our current conditions and in some of our political history, traditions, and institutions. The US Declaration of Independence is a document that lays a foundation that seems to me to be far superior to many alternatives that one can observe elsewhere in time and space. With its emphasis on individual liberty and consent, and its expression of the wisdom for people in general to alter or abolish a government that interferes too much in their lives, it comes near my own perspective.

However, today I’d like to make a few brief critical comments on the Declaration.

Here is an excerpt from the Declaration of Independence of the original thirteen united States of America, containing the main philosophical content of the document:

We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty, and the pursuit of Happiness, -That to secure these rights, Governments are instituted among Men, deriving their just powers from the consent of the governed, —That whenever any Form of Government becomes destructive of these ends, it is the Right of the People to alter or to abolish it, and to institute new Government, laying its foundation on such principles and organizing its powers in such form, as to them shall seem most likely to effect their Safety and Happiness.

Critical Comments:

Few truths — perhaps none — are universally self-evident.

Humans are not “created equal”, whatever that means; they are diverse and different in many ways, in body, mind, environment, interconnection, history, ability, skill, et cetera. Perhaps the only thing that is truly equal or the same amongst humans is the basic laws or patterns of physical reality they live in, including probably some patterns of physiology and psychology for survival, satisfaction, and happiness. Perhaps we can construct some political principles that would be wise to apply uniformly across humanity, but that would be an ideal to strive toward, not a political reality.

Without defining what a “Right” is, it is a kind of “magic” word that leads to sloppy thinking and inability to problem-solve effectively. This sort of thing can lead and has led to political confusion, turmoil, disaster, and mass slaughter.

In my view, reality seems to be much more complicated than the concepts and narratives we use to describe it. The same goes for politics and political institutions. We cannot accurately make sweeping statements such as “Governments are instituted among people to secure certain universally self-evident unalienable rights.” Modern governments in technologically advanced regions seem to be extremely complex institutions with a wide variety of purposes and motivating forces interwoven with each other, stretching over a deep history into the present. This complexity doesn’t mean that these governments are finely-tuned and good solutions to our political problems — I believe the complexity arises for both good and bad reasons (resulting from freedom and innovation as well as philosophical contradiction and political deals, exploitation, and some legitimate compromise). Better solutions may at once be both simpler and more complex, in different ways.

To improve upon the social and political systems we have today, we need to be able to confront this complexity humbly but with realistic optimism and ambition. We need to recognize the benefits we accrue and the successes we obtain with the current structures as well as the derived injuries, failings, and destruction. We need to recognize the sources of the good and bad outcomes, the alternatives we have, and the risks and rewards we face in our attempts to improve and innovate.

The following claims may or may not be true, but they’ve become intuitive to me and have been useful models for my own understanding of self.

Key take-aways

At the root of the self, approaching a perspective that could be called “pure awareness”, there is so little “self” left that one may consider it non-self, or one may alternatively consider it to be true self or core self. The distinction between “non-self” and “true self” is not necessarily very helpful or meaningful, and so one may declare a kind of conceptual “non-duality of self” at this low-level core of one’s experience.

The “self” is a kind of “scale-dependent” idea; the self is very real at certain scales, but the concept becomes less useful at the more fine scales of experience. One could say that the self is not really an illusion, but that the sense that one’s idiosyncratic personal self is at the core of one’s experience and being *is* an illusion. That higher-level self is real but not fundamental.

One critical aspect of the non-dual or non-self perspective is that one’s will, which is a critical aspect of higher-level self, is viscerally observed to be “externally” determined and outside of the core self of pure awareness.

(I think some of my writing here is a bit sloppy, since I’m not always talking about awareness itself but a particular experience when sufficiently detached-from and non-absorbed-by the higher-level processes/perceptions of self. Keep that in mind.)

Context

One form of wisdom is self-knowledge. Misconceptions about the self in general and oneself in particular can be part of a network of unhelpful or false thoughts that perpetuate a tremendous amount of unnecessary mental suffering and unwise action. Self-knowledge can potentially bring more effective behavior, better relationships, and greater peace and self-acceptance.

Knowledge of oneself can be gained in many ways, including seeking out a variety of experiences alone and with other people, along with careful observation of one’s own feelings, thoughts, and behavior over time to notice patterns and discover better mental models of oneself and the world. But one important avenue for exploration is to examine the meaning of the concept of “self”, and one’s concept of oneself, and see how the concepts inter-relate with one’s own experiences, both in the past and interactively as one considers “self-ness”.

In my own investigation of the concept of “self”, I’ve found that the self is not usually a well-defined idea; it is vague and can be analyzed and defined at several different levels. This vagueness and ill-defined-ness in general usage of the term leads people into confusion and blocks them from clear communication and understanding.

Levels of self

I see the concept and experience of “self” as many-layered and many-faceted; and I imagine a physical “self” as a many-layered and many-faceted physical process. At the highest, most superficial levels, I see the idea of self as incorporating all the things that make each of us as a person unique: our body, history, memories, personality, character, quirks, etc. But at the deeper levels, I see there are certain processes that are common to all of us, perhaps the most fundamental being the process of awareness itself.

Awareness can focus on many different kinds of things, and there can be many levels of “filters”. Some top-level filters are very personality- and condition-oriented, and will cause whatever is perceived to be interpreted within stories that can become helpfully or harmfully entrenched in our lives. Lower-level filters can take perceptions of something (say, light) and turn in from a collage of light and color into a layout of trees, rivers, mountains, etc — a landscape. These are perceptual and conceptual filters.

In meditation, or any time the mind is calmed enough to get out of the mental chatter and incessant thoughts, concerns, and emotions that tend to absorb the mind, one can have experiences from the perspective of a “deeper self” where fewer filters are being applied or are only partially being applied. Experiencing the deepest level of self, without the higher levels taking one’s immersed attention, can be explained as “losing the self” or experiencing “non-self” or “true self” (depending on one’s notion of “self”) and maybe a visceral realization of “non-duality of self” — or non-usefulness of the self/not-self distinction. This deep level of awareness can be seen as a kind of empty space in which the world appears, including the phenomena of the mind, which one ultimately has no control over.

The ideas of intention and control, of will and self-induced causality, make sense at the higher levels of self. But upon closer examination and detached observation, we see at the root of our awareness, the awareness has no intention or control. Perceptions are simply there and changing all the time. And we can examine the crucial question of how much control we have over our own intentions. At the higher levels, there are some kinds of control on some intentions, but at the lowest levels, we have no such control. Our intentions themselves are yet another arising and dissipating phenomenon in our field of perception.

This lowest-level perspective yields a visceral observation of the determinism of will and the rest of the changing, conditioned self. The observation and realization that will is not part of the core self, that it is determined and uncontrolled, can be shocking and enlightening. It seems to be a very low-level aspect of self that people presume to be at the core.

Non-Duality of Self

One’s awareness, or rather the contents of one’s awareness — one’s field of perceptions — seems to be unique and different from other people’s awarenesses. That uniqueness sets apart one’s awareness as essentially part of the “self” concept. Awareness, in this sense, has some self-ness. Awareness also seems to be completely necessary for a self to even exist. So awareness can count as a root of the “self”. However, awareness, considered as the lowest level self and root of the self, or “true self”, has very little of the qualities associated with the most broad conceptions of the self. One might even go so far as to say that awareness itself contains no self, especially given that it does not “contain” will. This rhetorical position is solidified by the conception of awareness as a kind of emptiness in which perceptions appear. Alternatively, one can even do away with the notion of awareness and simply “be” the perceptions. Boundaries and perceptual filters can dissolve and everything “becomes one”. Which of these perspectives makes most sense? Experiencing these things directly may lead to a notion that taking a solid stand any-which-way is not necessarily helpful. And trying to draw a dichotomy between self and not-self may not always be useful.

The concept of will is perfectly valid, but “free will” can be troublesome.

Will and Freedom

Will refers to desire, choice, intention, and determination. Inherent in the idea of will is some “actor” or “agent” that has, in some sense, a human-like internal experience of mind with which it experiences will, whether that experience is more or less rudimentary than a human’s. Such an agent can be said to have the property of will; the agent has will. We do not have extremely rigorous means of measuring will as of this writing; we generally observe behavior of an organism and compare it with our own behavior to infer whether something has will or not. Perhaps someday we’ll have a rigorous theory of physical mind instantiation (probably involving complex network dynamics) where we’ll be able to measure organisms’ bodies to reveal different degrees and kinds of mind, internal experience, and will. Or, another extremely unlikely hypothesis, given the evidence we have, would be that mind is separate from the body and somehow decisions get “beamed” to the body while the thinking and “willing” goes on somewhere else.

Freedom refers to a lack of constraint on some thing. Anything instantiated in reality must have some constraints in order to exist at all, so freedom is a relative term, referring to lower levels of constraint with respect to some higher level of constraint.

Free Will

What could it mean to have free will or freedom of will? And what does it mean to have a constrained will? It seems obvious to me that a (willful) mind instantiated in reality must be constrained and determined by whatever physical (or “non-physical”) processes are allowing the mind to exist in the first place. Perhaps freedom of will could refer to a relatively greater range of theoretically possible or experientially actualized desires held and choices made by one organism with respect to another organism. I think the common notion of “free will”, however, refers to the intuition that, given a specific choice made by someone, that someone *could have* made a different choice, and that person was “free” to choose among the possibilities. The person making the decision was not aware of any influence that “forced” the decision inevitably one way or another. However, this is completely indistinguishable from completely determined (“pre-destined”) behavior, where one’s desires and choices are a direct inevitable consequence of the events and processes that precede the choice. When the decision process itself is determined, there is no sense of external pressure. The feeling or actuality of “freedom” does not conflict with complete determinism in behavior. It is the ignorance of which decision is the inevitable one that leaves open the question of which of the available options will be chosen. This is a “compatibilist” position that proposes no contradiction between “free” will and behavior and determined will and behavior.

The word “free” here may simply be unnecessary and adding confusion to the concept. I tend to think and communicate in terms of “will” rather than “free will” whenever possible, since it may make more sense to think of a (predetermined) will that is, to some degree, unpredictable. The idea of will is sufficient. There is no need to postulate a counter-factual option that could have been chosen. There was, in actuality, a consideration of different options, and an option *was* chosen, even if that choice was determined and inevitable. No contradiction.

The concept of “freedom” seems to be more useful in the context of social and political freedom and lack of pressure, force, and threat from other beings, rather than some abstract, metaphysical, counter-factual, “absolute”, or impossible freedom.

Cases for Consideration

Let’s consider some more concrete cases to investigate will and freedom of will.

Consider four children. The first child is lacking in imagination and originality, easily swayed away from any self-directed behavior, and is very obedient, following instructions and suggestions with no resistance or consternation. The second child is more imaginative, passionate, independent, stubborn, determined, and disobedient — that is, more willful — yet thon seems unfazed and equanimous in struggles of will and when taking of punishment. The third is just as imaginative and passionate as the second, but very obedient, and willing to shut down thon’s own pursuits whenever asked or told to do so. The fourth is equally imaginative and passionate as the second, and very obedient like the third, but is tremendously anguished about putting off thon’s own pursuits when following requests and orders.

Which of these children’s wills is most free? Or is “freedom” even an appropriate metric to be used here? Would a better question be, which child has more will or stronger will than the others? Are these meaningful questions?

It seems that the latter three children have “free-er” wills in the “considered options” or “possible options” sense of freedom. They have more “powers of will”. It seems that the fourth child has the least “free” will in terms of being constrained and put-upon by social and/or physical pressure and force. Is the second or fourth child exerting the most “will power”? Which has the “stronger” will? Although the second child’s will seems to be “winning” more, the fourth child’s will seems to be fighting against a greater internal opposition. Perhaps a comparison is not really possible (without some rigorous science of mind and measurement of these children’s wills).

If a person desires to have a new desire (say, to crave healthier food) or get rid of a current desire (such as an addiction), some people can accomplish these goals through intelligent and persistent action. This is a particular kind of freedom, to shift desires and habits. But what separates those who manage to accomplish these goals and those who don’t manage? It seems that it comes down to a combination of inherent qualities of the person and sometimes (or perhaps always ultimately) unexplainable instances of follow-through versus non-follow-through. Even a generally conscientious person may have certain times when thon doesn’t follow through, and it can be difficult or impossible to figure out why there wasn’t follow-through in that particular instance.

Conclusion

Before I wrote this, I thought I might be able to show that “free will” is a contradiction and nonsense. However, I’ve shown to myself that there does seem to be some room for sense in the phrase, even if many people use the phrase in nonsensical ways. Still, I think in many cases it may help to replace the idea of “free will” with “will” and move on.

The magic of abstraction: “magic ellipses” creating infinity.

This post is a response to Steve Patterson’s article regarding the concept of infinity and its acceptance in the discipline of mathematics being what Steve calls “the greatest intellectual catastrophe of all time”:

• Cantor Was Wrong | There Are No Infinite Sets

Hi Steve.  As a friend, I thought I’d write up some thoughts to try to give you my perspective on this topic and give you more motivation to look more closely at my previous article on the same topic of infinity.

Regardless of your position on the metaphysical status of numbers, I claim that the idea of infinity and infinite sets is not logically contradictory, or at least that you haven’t demonstrated a logical contradiction.  To prove my claim I’ll have to show you what I see to be your errors.  I’ve already explained some of the errors and issues I’ve found in my previous article.

A definition for “infinite”: I prefer “limitless in some sense” over your definition “without inherent limitation”, since an infinite entity must have *some* inherent limitations.  (The example I gave in my previous article was an infinite ribbon which has an inherent limit on its width and thickness but not its length.  The set of natural numbers also has the inherent limit that the numbers must be the “natural” or “counting” numbers, rather than, say, fractional or octonion numbers.)

I’ll address specific claims you made in this new article.

Excerpt 1:

“What is the cardinality of the set of all even and odd integers together?” In other words, what is Aleph-null plus Aleph-null?

The answer: Aleph-null. The cardinalities are the same.

If this strikes you as logically contradictory, that’s because it is, but mathematicians have believed this for over a century.

This means they accept the following idea: a whole can be the same size as its constituent parts, because “Aleph-null” is the same size as “Aleph-null plus Aleph-null.”

They justify this by saying, “Regular finite logic doesn’t apply when talking about infinite things!”

There is no logical contradiction here, because if you add a limitless set to a limitless set, you get a limitless set (each of them being countably infinite in this case).  A similar unusual thing can apply to normal everyday objects like cohering water droplets: add one droplet to another droplet and (in this context) you are left with one droplet.  One plus one equals one.  Mathematics is contextual, and specific scenarios don’t have to match up with what you are already familiar with.  As long as definitions and context are sufficiently defined and understood, you can make sense of these “non-regular” cases.

Excerpt 2:

First of all, and most obviously, it’s a confusion about metaphysics. To ask, “How many positive integers are there?” is to presuppose an error. Sets aren’t “out there”. They are created. All sets are exactly as large as they’ve been created. There is no such thing as “all the positive integers”.

It’s like asking, “How many words does the largest sentence have in it?” And when you respond, “I don’t know, but at any given time, it’s a finite amount”, they say, “No! I can just add a word to it! It’s an actually-infinite sentence with an infinite number of words!”

In a certain sense, I agree with you that the sets that we reason with are created in the mind and are not necessarily also “out there” in the real world.  (So the “there” in the question could refer to “in imagination”.)  But we disagree about what qualifies as “created”.  As I explain in my previous article, you seem to have an overly and arbitrarily strict line between what you designate as “conceivable” and “inconceivable” (which we could translate here to “created” or “non-created”).  I would argue that, in a sense, you can “create” infinite sets in your mind through abstraction.  For most mathematicians, this is intuitive and uncontroversial.

So to ask, “How many positive integers are there?” is actually like asking “How many positive integers can you imagine?”  Well, I can imagine that there is a limitless amount of positive integers.  For any limit I come up with, I can exceed that limit.  In other words, there is a limitless quantity or an infinite amount of positive integers.  We could also say “there is an infinite number of positive integers”, if we generalize the concept of “number”.  Of course the number “infinity” is different from any natural number.  I see no contradictions here.

I would say this is less like asking “How many words does the largest sentence have in it?”, and more like asking “How many words does the largest string of words have in it?”.  One could answer this question by saying “As long as we allow infinitely long strings to be called ‘strings’, then we can imagine infinitely long strings of words, using a variety of combinations of words.  So there is no *one* largest string of words, but for the collection of largest strings of words, they all have a countably infinite number of words in them.”

Excerpt 3:

There is no “largest possible number.”

We’re in agreement here.  Even if we consider infinity as a number, there is always a larger infinity that you can create.

Excerpt 4:

The very meaning of “infinite” is mutually exclusive with the meaning of “set”.

A set explicitly means an actual, defined collection of elements. If you ever, at any point, have an actual collection of elements, you certainly do not have an infinite amount. In order to be collected, the amount must have boundaries around it – which is an explicit denial of infinitude.

There is no such thing as “actually-infinite amount”. What we mean by “amount” is precisely that it’s a finite amount. An “infinite amount” isn’t an amount at all. If infinity means “never-fully-encapsulated”, then it cannot be put into a set, by its very definition.

I disagree and already addressed this point in my previous article, but I’ll try to rephrase quickly here.  There are at least two issues.  One issue is how you define and interpret your definition of “set” (and whether you conflate “potentially existing-in-the-real-world sets” with “sets we can imagine”).  Another issue is where you draw the line for well-defined and “created” or conceivable or “collected” or “actual” elements. I would simply say that, for example, the set of all natural numbers *is* well-defined *and* conceivable in an abstract sense.  Hence it can qualify as a set, even with your definition of sets.

I’ve also addressed the issue of being careful with your definition of infinity.  I don’t think “never-fully-encapsulated” is a general definition for infinity and shouldn’t affect your assessment of the superior definition, “limitless in some sense”.  And if what we mean by “an amount” is precisely “a finite amount”, then we need to use a different word for infinite quantities; you can’t just arbitrarily define away the concept of infinite quantities.  I see no contradiction here.

Excerpt 5:

Universally, mathematicians will represent “the set of all positive integers” as {1, 2, 3, 4, 5, …} – implying that their set actually keeps extending into infinity.

I call these the “magic ellipses.” Somehow, if you throw three periods together, it allows you to complete an infinity – to say, “I am able to put boundaries around a sequence which has no end.”

This is a logical error.

The boundaries that are really being referred-to here are the boundaries of the definition of the set.  If, by the final curly brace, you understand what is being defined, then the concept of the set has been communicated successfully and you have “bound” the limitless set in your mind.  No logical error here.

Excerpt 6:

The infinity that lies between 0 and 1 is a larger infinity than “all the natural numbers”. This will strike most people as ridiculous, and that’s because it is.

It may seem ridiculous at first, but this claim is understandable after you put sufficient effort into understanding it.  Confusion does not always indicate that you’ve found an error; it may indicate that you have an opportunity to learn.

Excerpt 7:

Were I in academia, I would have to say polite and reserved things about my evaluation of Cantor’s argument.

I’d say this depends on the context; for instance, whether you’re a student or a professor, what your opinions and assertions are, and how you express them.  If you make arguments that are generally considered to be poor and unsound, you will find less opportunity within academia, especially in mathematics.  I think you are correct that your ideas about infinity are outside of the norm and most mathematicians would not be convinced by your arguments.  That doesn’t mean you’re wrong, of course.  Professional mathematicians are human and there is the possibility that large errors could persist (possibly even indefinitely) within the profession.

Excerpt 8:

“Infinite set” is a logically contradictory concept, no different than “square circle”, because it denies the law of identity – that A is A, or that “a thing is exactly what it is.”

“Infinity” is a denial of identity. It’s saying, “Never complete, never boundaried, never finite.” If something is identical with itself, then it certainly cannot be more than itself, which is precisely what infinity requires. If at any point, you’re dealing with Z, and Z is identical with itself, then Z is necessarily finite, as it cannot be more-than-itself.

You’re “moving the goal posts” again by changing the definition.  Something infinite need not be these things: “never complete, never boundaried, never finite”.  I discuss this in my previous article regarding different contexts and different definitions of infinity.  Infinity needn’t be “more than itself”.  Even if we take a (poor) definition of  infinity as “always bigger than”, and apply the trivial form of the law of identity (A is A), we simply get “always bigger than” is “always bigger than”.  We don’t get “A is always bigger than A” or anything like that.  I see no contradiction here.

Excerpt 9:

The correction is obvious: sets are generated by the human mind and are therefore finite. They are only as large as they’ve been created. By putting three periods together, one has not created an infinite anything. One has stopped thinking. Wherever the numbers stop, the numbers stop.

I could turn this around and say “By refusing to accept the concept of infinity, one has simply stopped thinking, and put up an artificial boundary for what is conceivable”.  I elaborate on this in my previous article.

Excerpt 10:

Numbers do not somehow stretch infinitely into the ether, with mathematicians vaguely pointing at them. Numbers don’t keep getting generated after you’ve stopped generating them – just like sentences don’t go on forever once you’ve stopped writing.

Actually, many things we discuss and think about are, in some senses, vague and never absolutely-fully specified.  We rely on our common experiences and intuitions a great deal, and I see no way to escape some level of vagueness.  I refer to this in my previous article when I mention “defining terms using words”.  Mathematicians have formally escaped the vagueness by taking the entity called “infinity” as an axiomatic element.  You are free to refrain from using it, but I see no contradiction in using it.

Excerpt 11:

Diagonalizing is simply a way to create a new number – one that is necessarily finite and only includes as many decimals as you’ve specified. Without “actual infinities”, Cantor’s entire project collapses on itself.

I think you get the picture by now that I disagree and his project doesn’t collapse.

Excerpt 12:

To be frank, if I were a mathematician, I would be embarrassed by the conceptual holes in Cantor’s argument.

Turning this around, there’s no need for you to be embarrassed by not understanding Cantor’s arguments.  It’s an opportunity for you to learn.  Or, if you can somehow convince me I’m wrong, I will be grateful and happy to concede that I was mistaken.  I sometimes try to figure out where I’m wrong, since I’m certain that I must be wrong in some ways, and if you can help me out in that regard that’s awesome.  On this topic, I think your chances are slim, but I enjoy your arguments and have been convinced by some of them in the past, so I’ll listen.

Excerpt 13:

Many contemporaries of Cantor mocked and despised his work.

People do this sometimes, even mathematicians.  Doesn’t mean Cantor was wrong.

Excerpt 14:

I have a hypothesis that I will write about more in the future. It has to do with sanity and mathematics – and the draw of the mentally-unstable into math.

I think there is something valid about this idea, but I won’t elaborate here either.

Excerpt 15:

It’s dangerous to uncritically use terms like “divide” when talking about mental conceptions. That’s a term borrowed from the physical world, where objects can be divided into their constituent parts – like a big ball of clay being split into two small balls of clay.

Numbers do not work like this.

I’d say they *can* and *do* work like this, if you represent them in an appropriate way, but this isn’t really an important point for my argument.

Conclusion

This article and my previous article may not fully convince you to change your position, but I hope they at least cause you to consider setting the bar higher for demonstrating logical contradictions.  And I hope you enjoy reading my arguments.

Photograph of galaxies and a galaxy cluster, with stars from our own galaxy in the foreground, by Bruce Pipes. Is it possible that these could be galaxies in an infinite universe?

This post is a response to Steve Patterson’s interview regarding the concept of infinity in mathematics and the real world:

• Ep. 8 – Completed Infinities: Possible or Not? | Dr. Gary McGuire

Hey Steve, great interview!  I like your questions and reasoning, and wanted to see if I could help sort things out.  I also enjoyed Gary’s responses.  So far, you are taking the position that an “infinity” or a “completed infinity” cannot be realized in the real world or universe, but I’d like to convince (both of) you that it is possible that they could be.

Summary of the problems I see in your reasoning:

1. misunderstanding the rigorous mathematician’s position
2. mistaking conceivability with possibility
3. having a binary view of conceivability (which could be too strict)
4. having a particular constructionist view of conceivable (which could be too strict and arbitrary)
5. having a constructionist view of infinity
   • (or assuming that the universe is necessarily conceivable in a strict sense)
6. misapplying a definition of “infinite” in a certain circumstance
7. mistaking “inconceivable in a certain sense” with “logically impossible”
8. missed equivocation in “infinite circle” argument (but caught apparent contradiction!)
9. missed equivocation in non-Euclidean geometry arguments (but caught apparent contradictions!)
10. convergence confusion (pie construction and Zeno’s motion/dichotomy paradox)

You’ll also find at least 7 points that I mention below that I agree with, along with enjoying your clear and critical reasoning.

Cautionary Preface: Note that a mathematical term or object doesn’t mean anything except to the extent that a definition and context is provided.  The looser the stipulation of the definition and context, the looser you can play with the term.  You can provide definitions and context by using words* or by demonstrating the usage of the term in practice.  One should also be aware that mathematical terms are sometimes the same as words used in casual colloquial speech, but one should be careful to keep the uses separate in your thinking if the definitions or context are different.

• *Defining terms using words can be troublesome because the definition’s words themselves require definitions, which contain more words, which require more definitions, et cetera, where all the words’ meanings are ultimately founded upon meaningless experiences that only gain meaning through our physical (biological, social) context.  Now that’s a whole other blog post waiting to happen!  But I imagine we’ll be able to get to some base layer of words that are clear enough for us to communicate effectively.  Here’s a relevant quote from John von Neumann: “When we talk mathematics, we may be discussing a secondary language built on the primary language of the nervous system.”¹See the Wikiquote John von Neumann page.

First I’ll address this (slightly rephrased) argument of yours:

For somebody that is not a mathematician, it can seem like a dicey way of reasoning, to think “let’s just take this [idea of sets of infinite size] as an axiom and work from there”, when it doesn’t seem like a clearly analyzed possibility.  So if we were to go outside of mathematics and one was to say “I believe in X”, and someone else says “why do you believe in X?”, and the one says “Well, I take it as an assumption and work from there”, we wouldn’t accept that as being a satisfactory reason.  Surely there must be more justification than just taking it as an assumption.  Do you think that’s fair?

I think you have a good point, but you’re misunderstanding the (rigorous) mathematician’s case somewhat here.  The mathematician doesn’t generally say “I believe in X”; the mathematician usually says “if we suppose X, then what follows is…” or, more extremely, “X seems conceivable enough” so “if X is not shown to be logically impossible, I feel comfortable in using it to develop a mathematical line of reasoning”.  In any case, the mathematician would say that “if X is later shown to be logically impossible, then everything that follows in this reasoning will not be logical”.  Now, beyond that, what’s interesting is that, for the mathematician, being illogical is not necessarily a fatal flaw, as long as you can find interesting non-trivial patterns in various kinds of illogical structures.  To a mathematician, the term “logical” can simply be another mathematical term, divorced from colloquial language and free from normative value.  If you want to apply these structures to your own reasoning in the real world, however, the translations back into real-world meaning and consequences will be important.

I think your point is essentially that if we want to be certain in our conclusions, then we have to be certain in our axioms or assumptions, and every step of our reasoning.  I agree with that.  But what we’re trying to get certainty about is whether a specific idea is possible or not, or, rather, whether it’s *conceivable* or not.  (I actually believe that the universe could be inconceivable, but I act as if it is conceivable to a certain degree because that operating assumption has shown itself to be productive and I can’t imagine making any progress otherwise.  So for me, conceivability is not the same thing as possibility.  But for the sake of argument, let’s take them as the same for now.)  You seem to think that conceivability is a binary thing: either it is or it isn’t.  But I’m not so sure.  To me, it seems that conceivability is more like a continuum.  For example, the number 2 is very conceivable.  I have direct experience with it and great understanding of that quantity.  But a trillion?  Or 6.02 x 10^23?  These numbers are so large that it’s arguable whether they are truly conceivable in the same way.  They are conceivable as relative quantities, where you can do a sort of mental ladder of multiplication, and get an abstract understanding.  But can you visualize 6.02 x 10^23 things?  No, absolutely not.  Now, for the case of an infinitely long line, in your imagination you can have a kind of abstract visualization of it.  Can you really wrap your mind around that kind of extent?  No.  We have no direct experience with these kinds of things, but, still, our minds can get a kind of abstract conception of them.  So I would say that *in a sense*, all these things are conceivable.

Also, I think that you are thinking of “conceivability” in a particular “constructionist” or “constructivist” mindset, which is not the only way to think about conceivability and may be too strict.  What do I mean by constructionist?  I’ll give you examples.  Suppose we take a definition of a “set” to be a “collection of well-defined and distinct objects”.  (Note that we’re using words that need defining.)  By “distinct objects” I basically mean no double-counting objects.  Here are two constructionist conditions placed on the idea and definition of a set:

1. The Super-Extreme Constructionist Set:
   • “Well-defined” means actually conceived in your mind at some point in the past and intuitively understood in your experience, and “collection” means stated explicitly, either mentally, vocally, or in writing, together in sequence.
   • If these conditions are not met, then the proposed “set” is not a valid concept; it is “inconceived“, so no proof has been given of conceivability, and therefore it should not be admitted as conceivable.
2. The Less-Extreme Constructionist Set:
   • “Well-defined” means you can imagine the physical possibility (even if you’d need to live a very long time) of taking time to understand each element in question with respect to other elements that you intuitively understand in your experience, and “collection” means either explicitly stating the members or stating an intuitive instruction for picking, finding, or generating each element, and this instruction must be physically possible to execute (again, even if you’d need to live a very long, but finite, time).
   • If these conditions are not met, then the proposed “set” is not a valid concept; it is not theoretically physically mentally constructible, and therefore it should not be admitted as conceivable.

In both of these frameworks, a set cannot have an infinite number of elements, because one could never physically think of an infinite number of objects.  But two constructionists of these sorts would disagree about whether the set of integers from 1 to 5 billion is a valid concept, since neither has ever sat down and thought about all those numbers.  It could even be argued that the latter constructionist could say that the set of integers from 1 to 6.02 x 10^23 is not a valid concept because, as I said above, it is questionable that any human can *really* intuitively understand the quantity 6.02 x 10^23.

Where to draw the line in this kind of continuum of constructionism is not clear.  This is essentially what I was saying above about the continuum of conceivability.  You can “pound your fist” about where it is appropriate to draw the line, but can you make a persuasive argument?  My argument would be that there is no need to draw a line, but you can draw a personal line, if you like.  You may find that, practically speaking, you reach the same conclusions as everyone else (as the “finitist” schools reach the same conclusions as the “non-finitist” schools).  But this seems to be a matter of taste, not principle.  As you ascend up the continuum of conceivability, not just to infinity but to higher orders of infinity, another John von Neumann quote becomes relevant: “In mathematics, you don’t understand things, you just get used to them.”²The Oxford Dictionary of American Quotations. Edited by Hugh Rawson and Margaret Miner. Page 601.

Now, I agree with your various definitions of infinite (never-ending, or without boundaries, or limitless, or never-completed) as being common mathematical (and colloquial) definitions of infinite, and to this list we could add “unmeasurably great” and perhaps more phrases like “impossible-to-traverse”.  But in different contexts, or with different interpretations, these definitions can have different meanings and apply or not apply in different ways.  For example, an infinitely long ribbon of a certain uniform width and thickness, or “infinite ribbon”, would not be limitless in every sense.  It’s not limitless, period.  It’s limitless in length, but limited in width and thickness.  It’s also not “never completed”, since that definition assumes a process, which is not necessary for an infinite ribbon to exist.  It could have simply always existed or been created by a process of infinite extent but finite time.  The same goes for an infinite set, if such a thing exists in conception or in the real world.  It is not “never completed”, unless you’re talking about the process of trying to count or collect or individually consider each element, which for an immortal yet physical being would be a never-ending process.  So infinite doesn’t always imply “never completed”.

A problem in your argument arises in your definition of a (mathematical) set, and your interpretation of that definition, possibly along with a conflation of it with a (colloquial) set of things in the universe.  You give a definition of “set” that is “a definite collection of elements”, and then ask “how could you have a definite collection that is never-ending or never-completed?”.  It seems that you are assuming that “definite” here means or implies “finite”, but that is not necessarily the case.  “Definite” could merely mean well-defined and understood or understandable in some sense.  The set of positive integers is “definite” in which numbers are stipulated.  Of course, if you do take a mathematical “set” to be finite by definition, that does not limit colloquial sets of things to be finite.  However, I happen to think your definition of “set” is pretty close to a colloquial definition.  There’s just nothing that leads me to think that it can’t be infinite in size.

If I’m characterizing your thought correctly, I agree with you that if we analyze our colloquial understanding of the concept “set”, and if we find that the idea of an infinite set is logically contradictory, then we cannot reasonably speak colloquially about our universe containing an infinite set of things.  It would not be rational and it would corrupt our reasoning about our universe.  I agree with you on that.  However, I don’t think you’ve demonstrated such a logical contradiction.  (You’ve shown other contradictions, which I’ll address.)  I think, at most, you’ve effectively merely argued that an infinite set is “inconceivable” in a certain sense.  And I distinguish “inconceivable in a certain sense” with “logically impossible”.

To illustrate this distinction further than I already have above, here’s a question with several possible answers.

Question:

• Q)  Does the universe have a beginning of time?  (A first moment?)

Some Possible Answers:

• A0)  The question is wrong.
• A1)  Yes.
• A2)  No.
• A3)  It depends.
• A4)  The question is vague and can’t be answered.
• A5)  The question is vague: Yes and No.
• A6)  Yes and No, absolutely.
• A7)  You and your silly words; they will get you nowhere.  (Spoken by someone with a deep ineffable understanding of the universe.)

Even if we limit the possible answers to A1 or A2, I propose that any answer to this question is inconceivable in a certain sense.  Considering just A1 and A2, I both cannot conceive of a beginning to time, with a moment that has no past, and cannot conceive of infinite time.  They are both mind-blowing and beyond my human experience and intuition.  However, I can, in a kind of abstract, removed sense, consider these options as valid possibilities.  I can envision them in a loose way, and in that sense they are conceivable.  In the prior sense, these are inconceivable possibilities (A1 and A2), but I see no reason for them to be logically impossible.  (There is a chance, though, that with better understanding, I will someday find a resolution to this question that is in no sense inconceivable, and it may have more to do with my state of mind or awareness than my argumentation.)

In a similar way, an infinite set is inconceivable in a certain way but conceivable in a looser, more abstract way.  I see no logical contradiction.

You did bring up some apparent contradictions, though, that I should try to clear up.  What about the “infinite circle”?  What do people mean when they say “consider a line to be a circle with infinite radius”?  What they’re really doing is generalizing the word “circle”.  They are changing the meaning and scope of “circle” without saying that’s what they’re doing.  What they’re probably doing is no longer using a simple geometric definition for “circle” but rather using an algebraic definition, like x^2 + (y-r)^2 = r^2, and equating the limit of this as r goes ever higher to the case where “r is infinite”.  In doing so, they admit a circle with zero curvature (which naively seems like a contradiction).  It’s a matter of semantics or convenience whether you accept the line as a kind of “circle” or not, but it’s something to be wary of to keep your thinking clear.  I am happy to see your demonstration of clear thinking!

As an aside, note that humans often leave important steps and assumptions unsaid (or even unconscious) when they use words and make arguments, so that even in the exceedingly rare cases where they (may be) correct, they cannot adequately explain why.  Even mathematicians, who are some of the most precise, literal, and expository people on the planet, leave things unsaid sometimes (often with the assumption that other professional mathematicians will be aware of or familiar with the missing steps).  It’s even more the case with non-mathematician mathematics educators or popular expositors, who may not be aware of the full detail of the ideas they are explaining or may want to keep the argument short for their uncommitted audience.  Plus, if you really tried to spell out everything in complete detail, you may not live long enough to complete the process.  So I’m not surprised that you’ve not gotten a satisfactory explanation of these concepts yet.

I think the issue with non-Euclidean geometry is the same.  Just as people abuse the term “circle”, they abuse the term “straight line”.  They really mean geodesic, which is a path that is traversed when one is *attempting* to move in a straight line.  This is a kind of “straight”, and it forms a kind of (possibly curved) “line”, so this usage is somewhat understandable.  To find out whether your geodesic actually is a Euclidean straight line requires investigation.  Also, given your understanding of the word “triangle”, it’s completely valid for you to distinguish between triangles in flat planes and “things with three angles and three vertices formed by closed path of three geodesics”.  Excellent distinction there!  An interesting question is, if you find yourself in a space with curvature, is it necessary that your space is embedded in a Euclidean super-space?  Intuitively, it seems necessary, but is that really true?  Perhaps, topologically, it’s not necessary.

Finally, you discuss convergence and limits in the context of two apparent paradoxes.  The first specific example you give is the construction of a pie by adding 1/2 of the pie, then 1/4 of the pie, then 1/8, 1/16, 1/32, and so on.  Do you ever reach the end of the construction, where you have the whole pie in place?  Your stance is that you will never have a completed pie.  Well, there are a number of ways of looking at this scenario.  We could say, imagine doing this in the real world, perhaps where the construction is done by robots over possibly a very long time.  At some point, we’d reach a fraction of the pie that is so tiny that it would yield a fraction of an atom.  At that point, you could claim that you are done (since atoms are coming and going all the time), or you could just throw the last atom on.  In other words, this scenario is not completely realistic, given a pie that’s made of atoms.  You can’t get arbitrarily small fractions of a real pie that will truly add more “pie”.  (This conclusion doesn’t negate the concept of an infinity in the real world, however.)  Another way of looking at this is in an idealized non-real world where you can perform un-physical processes.  In such a world, you can add each fraction of the pie at a fraction of some period T that decreases at the same rate as the fraction of the pie added.  So, say T = 1 minute.  Then you add 1/2 of the pie during 1/2 minute, 1/4 of the pie in 1/4 minute, and so on.  Thus you complete the pie construction in 1 minute.  Of course, if you allow arbitrarily small fractions of pie, but you do not allow arbitrarily small fractions of time for performing each addition, then the process will take “infinitely long” and you will never complete the pie construction.  You will only get arbitrarily close; by this I mean, you can choose however far in the future to look to see the pie, and so you can see the pie at any nearly completed stage you choose, including “inconceivably” close.

As for Zeno’s motion/dichotomy paradox, where instead of looking at pie construction, we’re looking at a runner traversing a distance d, a similar reasoning applies.  We don’t know for certain whether space or time is actually discrete in the real world, so we can conceivably chop the motion by space *and* time — not physically but theoretically.  So the runner could still cover the distance d in finite time, even in a universe with continuous space and time.  As you mention, space could be discrete, and I agree with you that in some sense this is more intuitive, but it’s also inconceivable in another sense: motion becomes more “spooky” where things “hop” from one point to the next.  How do they do *that*.  What *is* a point anyway?  Really, either scenario (continuous or discrete space) is mind-blowing to me.  In fact, I think any scenario or model you propose will be in some way inconceivable and mind-boggling if you look at it deeply enough and will ultimately rely on some assumed unprovable axiom(s).  (The point is to find the most useful axioms.)  Again, though, either of these discrete or continuous scenarios does not negate the existence of, say, infinitely large sets in the real world.

So, in conclusion, I’d like to summarize the important points:

1. In a sense, perhaps any claim is inconceivable, relying on mind-boggling axioms.
2. Infinities are conceivable, although I grant they are higher in the continuum of conceivability.
3. Even if infinities are not considered strictly conceivable or constructible (by our minds or finite processes) they are possible without logical contradiction.
4. Discrete space or time or matter or energy does not rule out the possibility of infinities either.

Beautiful mathematics: a section of the Mandelbrot set. (Image taken from the Wikipedia article on this set.)

Short answer, phrased in three ways:

1. The science and logic of patterns.
2. A pattern club and tool box.
3. The game of abstraction.

Long answer:

What follows is a fairly loose description of mathematics without much support given.  Maybe later I could fill these thoughts out more to address some of the deeper issues discussed, for example, in the Stanford Encyclopedia of Philosophy article on the Philosophy of Mathematics.

I’d say that math is the science and logic of patterns.  It is inspired by what we observe in the real world but is ultimately based upon what we can imagine and the implications of our imaginings through our reasoning.  The patterns can be in space, or time, or sequence, or any realm you can imagine, including the patterns of logic or reasoning itself.  You can do mathematics by recognizing, creating, or imagining patterns, categorizing them, finding the relationships between and within them, creating language to describe all this, and making your concepts and language more precise and exact.  If your results are sensible enough, they may be accepted by professional mathematicians as qualifying as mathematics.

(A few notes and caveats:  You will probably have to use a certain amount of commonly accepted mathematical language or jargon before anyone will pay attention to your work, since it takes a lot of effort to translate idiosyncratic expressions into familiar ones.  Also, it seems exceedingly rare, from my knowledge, for something to be accepted as rigorous mathematics and later to be rejected, so the bar for “sensible” has historically been quite high.  Furthermore, there is a chance that your work will not be recognized as correct until after your death, so you may be ahead of your time.  And, apparently, your work doesn’t even have to be consistent to be accepted nowadays, but it probably has to be “paraconsistent”.)

Amazingly, if you take interest in this kind of activity and get into the swing of doing it and expressing creativity, you can spend a lifetime generating new mathematical concepts and honing and tweaking existing ones.  The richness of mathematics just keeps growing as the years pass.

Mathematics can be done alone, by a single person, but people tend to interact and so mathematics tends to be shared too.  Taken cumulatively across the world, mathematics is a global social enterprise and a growing collection of concepts, procedures, tricks, and arguments for reasoning about patterns — a kind of pattern club and tool box.  Learning mathematics can also be more enlightening when learning the history, lore, myths, people, and context of mathematical development.  So mathematics comes with a collection of stories as well.  As a human endeavor it has a colorful history with the full range of emotion that comes with any human endeavor.

Perhaps even more amazingly, as mathematics has grown from its crude and physical beginnings and become more abstract and esoteric, it has frequently and nearly continuously helped to advance the human pursuit of the physical sciences, to understand and manipulate the world around us.  Granted, it seems necessary that the surface-level patterns of our world must be fairly understandable, because otherwise how could humans have evolved to have the capacity for understanding in the first place?  But it could have been the case that the underlying patterns in our world were extremely complex and required tremendous progress in mathematics before much progress could be made in science.  It’s possible that we do encounter such ruts now and then, or perhaps we are even in such a rut right now, but it is impressive how much progress has been made in the physical sciences with the use of mathematics.  This “effectiveness” of mathematics has sometimes even been called unreasonable.  I should note too that advances in physical science as well as computer science and other fields have provided inspiration for advances in mathematics, so there is a kind of virtuous symbiosis amongst these fields.

Does math ever lead people astray?  I would say not the math itself, but its misapplication can.  (I believe more errors are made in various branches of science, and it is more likely that “science” leads people astray sometimes.)  Perhaps you’ve heard the phrase, “There are three kinds of lies: lies, damned lies, and statistics.”  Erroneous reasoning wrapped up in mathematical jargon can seem much more respectable and convincing than the reasoning stripped bare.  Mathematics can help to give legitimacy to a bogus economic theory as it supports a deceptive political agenda.  A similar thing happens when science is misapplied or mimicked to support pseudoscientific beliefs.  But, hopefully, those with good mathematical, scientific, and philosophical understanding can help to publicly expose and discredit such theories and beliefs.

I would say that math, considered separately from its applications, is the game of abstraction.  If you dream up a pattern of abstraction to play with, then you are doing mathematics (even if as an amateur), and there’s a ridiculously significant chance that even if it seems useless, some day, someone will find your game a useful one.  (Or, maybe it’s already been dreamed up and used!)  Mathematics is a joy in itself, but combined with its applications it is a source of great power and effectiveness.  Let’s have fun and use it wisely.