For me, I believe mathematical objects, as well as mathematical truths, exist. So I'm a Platonist.

This is by analogy with the intuition that material objects exist. These exist in the sense that they can be imaged, measured and perceived by many people who will get the same measurements of the same things, at least within a fine tolerance that would improve as our measuring instruments improve.

Similarly, mathematical objects and truths can be observed and measured by many people, or even software, and multiple computer programs (assumed to be debugged) and mathematicians (assumed competent) will observe the same truths about these objects.

This perception of material objects, or mathematical truths, doesn't need to occur in order for the things at issue to exist; it just needs to be in principle possible.

Let me add that I believe that if any hypothetical beings, or even high-powered computers, were acting like mathematicians in distant parts of the universe, they would eventually come upon the very same mathematical truths that mathematicians on Earth would, perhaps in a different sequence. Math is like a landscape waiting to be discovered, no matter where the discoverers happen to be.

Well, to say "The cat is on the mat" means I'm thinking of a certain thing that is a cat, and of another thing that is a mat, and the first thing is on the second thing."

But my point was that this kind of statement is slippery since both cats and mats are not clearly defined.

Sure, math is a social enterprise. But that's not all it is.

As I see it, mathematical truth exists independent of whether there are any conscious beings who know about it.

A thing has mathematical existence when a preponderance of working professional mathematicians say it does. — fishfry

So are you saying that when Georg Cantor first defined infinite sets ca. 1871 and there was great resistance among the world's mathematicians, infinity didn't exist yet?

What was the question again? Truth? Who said anything about being able to "describe the exact physical characteristics" of the cat or the mat?

Are we talking about how to communicate conveniently, or are we discussing what truth consists of? I thought it was the latter.

Well, no, the number of distinct digital photos of a given resolution is finite. — SophistiCat

Okay, if there is a fixed computer we're using.

But if the photo is say K x L pixels, and each pixel contains N bits of information, then by increasing N (to represent hypothetical better and better computers) then the number of K x L photos that can be conceived is infinite.

What is "the cat"? What is "the mat"?

Most of the time, we know what these words mean. But what about (let's take the mat) when the mat is being fabricated?

If we don't know exactly when the mat-in-process-of-being-manufactured is in fact a mat, then we don't *really* know what the mat is. (Same thing with a cat as it's being biologically conceived and developing in the womb, or as it is dying, let's hope at a ripe old age.) Or the mat when it is falling apart eventually.

Cats and mats have not only spatial but also temporal extents ... but we don't know what those extents are. (Or as the cat is digesting and assimilating its food, when exactly does the food become the cat?)

The fact that these questions have no clear answers means that the truth (or not) of a simple statement like "The cat is on the mat" is much less clear than it may first appear.

the notion of vertical and horizontal Euclidean lengths is incommensurate with the notion of diagonal Euclidean lengths — sime

What does that mean, to say that the notion of vertical and horizontal lengths is incommensurate with the notion of diagonal lengths?

It is somewhat subtle to define for which subsets of the plane the concept of length makes sense, and to define exactly what the length is of such a subset. But in the standard definition, there is no distinction between diagonal lengths and horizontal or vertical ones.

If we are constantly changing our opinions as to the facts of the past on the basis of new information, then why should we believe that the past is real and immutable? — sime

You ask why the past is immutable. But as a reason not to think of it that way, you mention only that our thoughts about the past can change. There is an important distinction to be drawn between a) a thing (the past), and b) our thoughts about it later.

No, if the arc length decreases any faster than in the examples that have been considered so far, it will converge to the length of the diagonal, as we intuitively expect. — SophistiCat

I wasn't explicit enough by what I meant by"scaled down versions".

First of all, I'm switching from the diagonal to just continuous functions defined on some interval of the real line. Same picture just viewed from another angle, but easier to talk about.

Now, what I mean by a "scaled down" is the graph of a continuous function y = f(x) that has been modified so that all linear measurements of the whole graph are shrunk by the same factor C > 1.

So instead of y = f(x) we're now looking at y = (1/C)*f(Cx).

This is essentially the same way the original stairstep pattern is scaled as it gets closer to the diagonal line. So if the graph is a bunch of semicircles of radius = 1, end to end resting on the x-axis, the arclength above a diameter like say 0 <= x <= 2 will be π. The ratio of arclength to the length of the interval on the x-axis is π/2.

If we shrink the whole graph down by a factor of say 100, then the semicircles now have radius = 1/100 and the arclength above a diameter, say the interval 0 <= x <= 1/50, will now be π/100 and the ratio is again π/100 / (1/50) = π/2.

If you believe in free will, the future is not determined. — Metaphysician Undercover

I believe the future is determined regardless of what one believes.

e cannot make truthful 'is' and 'is not' statements concerning the future because it is indeterminate, characterized by possibility. — Metaphysician Undercover

I would say that this is true relative to human understanding (only) — but not in an absolute sense. Because humans experience time one instant at a time and don't know the future with any certainty (needless to say).

But in an absolute sense, the future is just like the past or the present — something will have happened (regardless of whether we know what that is now).

For any continuous function like whose arclength for a <= x <= b is greater than b-a, its scaled down versions will still have the same ratio of arclength to b-a. So just about any continuous function at all that's not a constant.

Maybe I've misread, but you appear to have "ultimate truth"="truth." What meaning if any attaches to the word ultimate in this context? — tim wood

As I tried to suggest in the original post, I'm *not* considering facts like "The book is on the table," no matter how clear and useful they may be for us humans — because nouns like "book" and "table" aren't well-defined in an absolute sense. (They belong to fuzzy sets, especially at the boundaries.)

Maybe you mean the future is indeterminate with respect to the present. But (and I'm not considering the "many-worlds" hypothesis of multiple futures here) the future will occur, regardless of whether anyone knows what it consists of now. And when it does, *something* will happen.

I'm not speaking of what humans can or do know. Only what's true. And as they say, que sera, sera.

I'm afraid I don't know what makes an axiom appropriate, but also I don't know what an axiom without a verb means.

Cogito ergo sum does establish that "Thought" exists now: There is thought now, is how I like to generalize that aphorism. However the "I" seems like an add-on. — Pantagruel

I agree that the "I" portion of "Cogito, ergo sum" is gratuitous and very weak.

Specifically: If the conclusion is that "I exist", then how can "I" be used in the premise, since it was not yet established that "I" exists!

What I'm able to extract from that famous quotation is this:

"Because there are experiences, something exists."

(Thoughts are just one type of experience, so I see no reason to limit the interpretation to thoughts alone.)

There are at least three distinct meanings for the word "color": 1) the type of visual experience we can have. 2) the property of a material object that causes this type of experience. 3) a ray of light at a certain frequency.

There are some problems with both 2) and 3).

With 2): The color that a material object appears to someone looking at it depends on the ambient light reflecting off it. (You may have been surprised after buying clothing in a store and then going outside and noticing it seems to be a different color in sunlight.) In addition, even one person's right eye and left eye can perceive color differently at the same time. As well as the Land effect, which is that color perception also depends on on the perception of colors surrounding something you are looking at. And so, the "color" of a material object depends on some things besides the object itself.

And with 3): Although physicists understand that it is the frequencies of light rays entering the eye that are the main determiners of how that light is perceived, there are actually infinitely many combinations of light rays that give rise to the same color perception. (The exceptions to this are pure spectral colors.)

At the very least, in order to speak intelligently about colors it's important to say which definition is being referred to.

IvoryBlackBishop: Suppose you have a certain experience on a certain day. Then the fact that that experience was experienced is true, and nothing can ever change that fact.

This is completely independent of how anyone may choose to describe that experience.

When I call something fuzzy, it's not because human's concepts of it are fuzzy ... though that can certainly accompany fuzziness. It's because in an absolute sense the very thing itself is ill-defined, at least when you look at it closely. Consciousness — experiences that are experienced — is exactly what it is, no two ways about it. That's why I don't consider it fuzzy in the least.

Good questions.

1) The truth of physical reality means everything about elementary particles — of which everything is composed. (People have the same type of fuzziness that chairs do in that there's no agreement on exactly when a person begins to exist or stops existing.)

2) I maintain that consciousness per se is not fuzzy at all — it's only our attempts to describe it that are. (To answer whether redness exists would depend on what it means: Does redness mean the concept of electromagnetic waves in a certain range of frequencies? Actual electromagnetic waves? Or the experience of someone's seeing them?)

3) Mathematical axioms are not subject to being true or false. The things that are true or false are the facts that — from a given collection of axioms — certain conclusions can be logically derived (or not).
(There are some other things, too, that can be true or false regardless of the fact that they cannot be derived logically.)

In my opinion, consciousness does not emerge; it is present at all previous times. I believe that because I cannot imagine its emergence from a state where it previously did not exist.

Well, first off, it's not true. Science does not get closer and closer to some objective reality. — T Clark

On the contrary, I believe that that is exactly what science does. A claim in science is accepted only when the claim is verifiable, over and over, by others. Often this is a measurement of something. Measurements of things like the speed of light, the weight of atoms, the size of the electron have only gotten more and more precise as time goes on. Our knowledge of animal behavior and the classification of plants and animals only improves with time. Etc.That is the very definition of getting closer and closer to some objective reality.

There still is an indescribably huge collection of questions we don't know the answers to and, undoubtedly, don't even know enough to formulate the questions correctly. We very well may never know all the answers. But that doesn't change the fact that science is getting closer to knowing some objective reality.

I'm not referring to belief. By "ultimate truth" I mean truth that is not kinda, sorta, -ish.

That's why I mentioned how words are often fuzzy, to make it clear I'm not talking about fuzzy truth.

I doubt Tegmark's claim that "everything is mathematics" since, first of all, how could abstract mathematics per se generate any sort of physical reality? Seems preposterous to me.

And secondly, it seems even more preposterous that mathematics alone could give rise to consciousness, which we know is real because we experience it.

The word "paradox" has two meanings: 1) something that is true but self-contradictory, and 2) something that is true and seems self-contradictory, but in fact isn't.

This staircase paradox is of type 2). It seems contradictory because we expect that the length of the staircase ought to approach the length of the diagonal line that it approaches. It's not really contradictory, simply because that expectation is not right. (Mathematics is filled with many paradoxes of this kind.)

If a sequence C_1, C_2, C_3, ... of curves approaches a line L in such a way that, not only do the points of the curve approach the points of the line, but also the direction of the curve approaches the direction of the line, then the lengths of the C_n's will also approach the length of L. Otherwise it just isn't necessarily true, no matter how surprising this may be.

I don't "know" what you "mean".

That's the general idea, Dr. G., but to be exact I'm thinking of https://en.wikipedia.org/wiki/Natural_density .

Density and cardinality are different concepts.

What do you disagree with?

The definition of when two sets have the "same cardinality" is (as most people here know) when there is a bijection between the members of one set and the members of another set.

It's easy to prove beyond a shadow of a doubt that if you associate each integer N with the even integer 2N, that is a bijection between the set of integers and the set of even integers.

You may not like this definition; you may be unwilling to accept this definition (of being the same size). But you can't argue with the fact that there exists a bijection between Z and 2Z (the symbols mathematicians use for the integers and the even integers, respectively).

Someone pointed out that if you could select an integer at random (in some sense; this is very tricky to define!), then only 50% of the time would you select an even integer.

This is a valid point, but it doesn't affect the fact that Z and 2Z have the same cardinality.
What it does point out is the fact that the density of the even integers, in the integers, is only 1/2.

In case this interests you: Suppose X is a subset of the (let's say positive) integers. Let X_n mean the numbers in X that are no larger than the integer n. The fraction X_n / n tells what fraction of the first n positive integers — that is, the set {1, 2, 3, ..., n} — happens to be in the set X.

IF that fraction X_n / n approaches a limit (call it L) as n approaches infinity, THEN the set X is said have density equal to L in the positive integers. (It may be fun to try to find a set X of positive integers for which that fraction X_n / n does not approach a limit as n approaches infinity. There are many such examples.)

It “explains” why my socks and bubblegum are conscious, even though no one thought they were, but it doesn’t explain why the human brain is conscious the way the human brain is conscious, which is what we actually want to know. To put it mildly, panpsychism is irrelevant and pointless. — Zelebg

What I wrote certainly doesn't explain anything (in the sense of Chalmers's Hard Problem).

But it does suggest the scope of consciousness in the universe. Which may be of no interest to you, but to me that would tell me a lot about consciousness.

And my hypothesis that all matter and elementary particles possess consciousness does not really say that socks or bubble gum is conscious as a whole. What would be conscious as a whole are assemblages of matter that are highly self-interactive. Which is why I listed some things that qualify: plants, animals, viruses, molecules, atoms, elementary particles, clouds, planets, stars, galaxies. And this works hierarchically as well, so groups of people, flocks of birds, beehives, and even (on a yet higher level of organization) the United Nations are included.

Socks and bubble gum are not completely lacking in consciousness, either — because their components, at some level, have it. In these cases, probably just their molecules.

As for human consciousness, this hypothesis posits that, like all consciousness, it stems ultimately from some kind of nanoconsciousness at the tiniest level of magnification.

Thanks . Yes, for a long time I wondered what kind of arrangments of matter led to consciousness, and whether computer were conscious just because they can compute.

One thing nudging me toward going all-in on where consciousness occurs was watching recent time-lapse videos of plants. They "behave" just like animals except for being planted in one place (other than tumbleweeds).

Far simpler than what?

I've never seen an argument in favor of the claim that "consciousness is an illusion" that made any sense. If we're having any kind of illusion at all, we are having *some* experience, regardless of how it relates to physical reality. And the having of experiences is the definition of consciousness.

In math, cardinality is often expressed as an equivalence relation between two sets: Sets A and B are equivalent (in cardinality) exactly when there exists a one-to-one correspondence between them.
This is more primitive than the concept of number: Two children can confirm that they each have the same number of pieces of candy by pairing one child's pieces with the other's — which does not require knowing how to count.

In math, an infinite set is often defined by the condition that it has the same cardinality as some proper subset of itself (a subset that doesn't include all the members of the original set). And sure enough, the positive integers {1, 2, 3, ...} can be put in one-to-one correspondence with the set of even positive integers {2, 4, 6, ...} just by the formula n <—> 2n. (As everyone knows, you can't do this with a finite set.)

There is a very simple proof that shows that for any set X, the "set of all subsets of X" can never be in one-to-one correspondence with the original set X, no matter how big or small X may be. So the set of all subsets of the integers is strictly larger than the set of integers. And doing this again, we find that the set of all subsets of *that* set is larger than that set is. There is no limit to how many times this can be done.

But hold on to your hat! Because the operation of "taking the set of all subsets of a set" can be done infinitely many times! And the union of all the resulting sets is larger than any previous one. So there are actually infinitely many distinct infinities.

This suggests why there is no largest infinite set — because the set of its subsets is strictly larger.

I believe that there is no plant, animal, virus, molecule, atom, or elementary particle that does not have consciousness of some sort. As well as in the other direction: clouds, planets, stars, galaxies.

I doubt we have any idea of "what it's like" to "be" any of these things other than some animals. Consciousness is generated by the interaction of the universe with itself on an infinitesimal scale. This participates in a causal cascade — events leading to other events. The main things we usually assume are conscious are animals, in which all parts interact with each other constantly. But many other conglomerations of matter have the same property, clouds for instance.

I realize that these claims will sound utterly outlandish to most people. But the reason I believe them is that I find this to be the simplest explanation for what we know and observe.

I think the apparent flow of time that we experience can only be studied in relation to consciousness. All of space and time that has existed, exists, or will exist is part of reality. No events in the future can erase events that have occurred in the past. So everything that has happened, is happening, or will happen is part of what's true. Regardless of whether it's in the past, present, or future of a particular individual.

So as I see it, on the one hand we have the reality of space and time, or if you will, spacetime ... and on the other hand we have conscious beings who experience moving through spacetime along a curve that always has a positive velocity in the "time" direction. At least schematically.

The experience of the flow of time is a bit like how the wick of a candle burns down. It is this interaction of time and consciousness that seems to me the thing to focus on in order to try to understand what's going on.