You're entirely correct about that, but now you are understanding and agreeing with my point. If you throw a dart at the real line, you'll hit a noncomputable number -- a number whose binary or decimal expansion is random -- with probability 1. And you'll hit some computable number with probability zero. But there are plenty of computable numbers.

So yes, I backtracked. — noAxioms

Appreciate that. Just trying to carve out some clarity in a fuzzy discussion.

An infinite sequence is not 'some number' — noAxioms

Is 1/3 = .3333.... = 3/10 + 3/100 + 3/1000 + ... some number?

There's no difference in principle between decimal and binary notation. And binary notation is just a sequence of H/T choices. A real number between 0 and 1 is an infinite sequence of head/tail choices. If you are picking them randomly then the chance of any particular one is zero. But they all exist. That's the point. In infinite probability spaces, probability zero events may occur.

If so, the dup-Earth bit kinda falls apart, eh? — noAxioms

Well yeah, that's my point. You claimed there's a dup earth, I asked you to support your claim, and you have apparently backtracked. Yes?

No, disagree with this. A finite sequence has a nonzero probability. An infinite one is not a specific one, and has probability zero and does not happen.
Every particular sequence has half the probability of the flip sequence of one-less flip. That is not zero. Some probability-zero sequence is not a particular sequence and thus does not in fact occur. — noAxioms

You said all heads has probability zero hence can't happen. I pointed out that ANY specific infinite sequence has probability zero. Are you unclear on this point? By coding H as 1 and T as 0 and putting a binary point in front of the sequence, an infinite sequence of coin flips represents some real number in the unit interval. If you randomly pick one, the probability that you picked it was zero. But SOME number gets picked.

I didn't watch it either — tom

Well then I'm glad I didn't watch it! It's bad form to post a vid you didn't watch IMO. But now that you pointed us to the good parts I'll take a look when I get a chance.

I have no idea what you mean by "the last few years", — tom

Multiverse theory, string theory, speculative theories that have zero experimental basis and that can't be experimentally verified even in theory.

Indeed, it doesn't require infinite space. It (a type 1 world, not a duplicate) does at least require an expanding universe, else eventually light would have time to cross the distance. The dup-Earth requires space big enough to form duplicates of something, which could in theory be close enough to be visible from here once light had time to make the trip. — noAxioms

Correct. Even if the universe is finite there might well be type-1 multiverses, which are simply regions of space too far apart for light to have gotten from one end to the other since the beginning. Which is good, because contemporary physics holds that the universe is finite. There's no sensible theory of an infinite universe, since then set theory would become an experimental science. And there are no grant applications from physics postdocs proposing to investigate the Continuum hypothesis and the axiom of choice. That's because infinity is a mathematical concept and not a physical one. This could change tomorrow, but by contemporary physics, the universe is finite.

Edit: I think that mathematically, a coin cannot come up tails forever. There cannot not be a dup Earth given infinite space. The probability of of that is 0.000... which is zero. — noAxioms

In infinite probability spaces, probability zero events may still happen. Suppose you flip infinitely many coins and they come up in any sequence whatsoever: hthhthththththhthttthhthththt... say. A completely random sequence. What's the probability? Well, the prob that flip 1 is h is 1/2. The prob that flip 2 is t is 1/2. Etc. The prob of the first n flips being exactly what they are is 1/2^n, and that goes to zero as n goes to infinity. Every particular sequence has probability zero. Do you follow that point? All heads is just as likely as alternating heads and tails which is just as likely as the random sequence above. The probability is zero. Yet SOME probability zero sequence must occur.

Not sure I follow this. If there is a duplicate, there would seem to be an infinite number of them. — noAxioms

In your first paragraph you agreed that level-1 multiverses only need a finite universe. And now you are claiming an infinite universe. I ask you (again), what is your evidence for such a claim? Contemporary physics has no such theory other than pure speculation.

There is simply

a probability as to how far away (measured in non-linear units) the nearest one is. There are infinite type-1 worlds given infinite universe size. — noAxioms

Yes but the universe is finite as far as we know. So you're saying, "Well, if Santa Claus is real, he gets to every house in finite time on Christmas eve." That's true, simply by virtue of the premise being false. You say that such and so happens in an infinite universe, but your premise is contradicted by known physics.

In fact, each point in space is centered on such a world, so you are in a different one than I am. Light can in theory reach (immortal) you — noAxioms

So now there are immortal beings in an infinite universe? This is not the science fiction forum. What is the evidence for your premises that the universe is infinite AND that it contains immortal beings?

You're just making this up.

For type 1, the distant Earth is a true duplicate. The space is infinite, but the possible states in a finite space (say that of Earth) are not, so each state much eventually be duplicated given enough distance. — noAxioms

First, why is the space infinite? Your premise is that the universe is finite but sufficiently large that there are regions inaccessible to each other due to light not having had enough time to get from one place to the other. That doesn't require an infinite universe.

Secondly, even if the universe is infinite AND the possible states in a given region of space are finite, you STILL are not guaranteed a duplicate earth. Say there are two states, and infinitely many universes:

0, 1, 1, 1, 1, 1, 1, ...

So SOME state occurs infinitely often. But the 0 state is never duplicated. In other words if there are infinitely many universes, then SOME rock or bacteria or planet is duplicated for sure. But you personally, or the earth, are not necessarily duplicated.

Again though, you're betraying your original premise. You started with TWO regions of the universe that are causally and informationally isolated from one another. That's two, not infinitely many.

1: Places that are too distant to causally interact with here, ever. There is a duplicate Earth out there if you go far enough. — noAxioms

How does the second sentence follow from the first? Do the universes share the same history? Why should they do that?

For those of us who prefer not to sit through a video and who have only a nodding acquaintance with the topic, can you please remind us what a level 3 multiverse is?

I think it is time to take physics seriously. — tom

On the contrary, it seems that in the past few years it's becoming time to stop taking physics seriously. Theories that can not be experimentally verified or refuted are not science.

I don't understand what you mean by assigning the name "truth" to some property of an object, like pink. That doesn't make any sense to me.

Secondly, computers don't answer questions that they haven't been programmed to answer. They're not magic oracles. And Turing showed that there are some questions a computer can never answer. This result is in fact related to Gödel's result that sufficiently powerful formal systems can't resolve all propositions.

7.) Therefore there must be a purely actual actualizer of everything that exists. — darthbarracuda

Same old Craig cosmological argument. Same old First Mover argument. Same old false argument.

Consider the following model of causation. Let E(n) stand for Event n where n is an integer.

..., E(-4), E(-3), E(-2), E(-1), E(0), E(1), ...

Suppose each event is caused by the event immediately to its left. Then we have a model of events in which:

* Every event is caused; and

* There is no first cause.

In your version of this fallacious argument you have swapped in the word "actualized" for cause. So think of this model as saying that every event is "actualized by" the event immediately to its left. Now every event is actualized yet there is no un-actualized actualizer.

Your thesis is refuted.

By the way this example is known as ω* in the literature. ω (lower case Greek letter omega) is the order type of the positive integers. ω* is the reverse order. By extending the causality chain to the right, the model I presented is described as ω* + ω. Philosophers already know all about this.

by non-physical existence, theists mean something outside of the mind, don't they? — TheMadFool

I thought you meant that atheists only believe in the existence of physical things. If you didn't mean that at least I'll try to clarify my own thinking. There seem to be three levels of existence: things that are physical; things that are abstractions of the mind; and things that are outside the mind.

An atheist believes in the first two but not the third presumably, if by "outside the mind" you mean God. Or ... well, outside the mind is too strong. How about outside reason? A theist believes in God, and to have a belief is an act of mind. God may be outside the mind but faith is within the mind. And some theists accept that their experience of faith literally is their experience of God.

So this idea of outside the mind I'm finding a little tricky to get hold of. But outside reason makes sense. A theist would agree that although they have belief, it's a belief based on faith and not on reason.

Wieseltier, it seems, has been caught up in the slipstream of the Weinstein scandal — Wayfarer

Interesting. Even the great Searle has a harassment problem as you may know.

For atheists, existence means something physical - that which can be perceived through the senses and if you want to go the whole nine yards, something measurable.

In contrast to the above, existence for theists goes beyond the physical - beyond our senses and instruments. — TheMadFool

Don't atheists do math? Believe in justice? Follow the law? All abstractions. Non-physical.

Atheists believe in and use abstractions.

What would it take to change the global thought process? Is it even possible? Can a world function stably with out unification? — Unstable

Propaganda. Here is one of history's most brilliant experts in getting everyone to think the same way. Is this what you have in mind?

http://www.psywarrior.com/Goebbels.html

Are you sure that getting everyone to think the same way is a good idea?

Take Chang and Perrig on emergent algorithms regarding cluster formation — Akanthinos

Your quote from the developer of a particular software application claiming his program shows "emergence" is an example of what I'm talking about. A word with a vague and tenuous meaning, but that sounds impressive if you don't think about it too much.

Your example illustrates and supports my point.

But never mind that. I've given all this some thought and I retract my former statement that no AI shows emergence.

Rather, I say that pretty much everything shows emergence, from simple programs to the world around us. By itself, emergence is a vaguely defined term that conveys little or no useful meaning.

Wikipedia defines emergence as

In philosophy, systems theory, science, and art, emergence is a phenomenon whereby larger entities arise through interactions among smaller or simpler entities such that the larger entities exhibit properties the smaller/simpler entities do not exhibit.

https://en.wikipedia.org/wiki/Emergence

There's a more detailed and nuanced definition or set of definitions in the main part of the article, but this will do for our purposes.

Well, what's emergent then? A glass of common tap water is emergent. Hydrogen's a gas, oxygen's a gas, put them together and it makes water. Water is wet. Hydrogen and oxygen "self-organize," a phrase used in the article, into something that has properties neither of the ingredients have.

That's emergence.

Speaking of ingredients, how about some nice light fluffy scrambled eggs? Eggs by themselves are not fluffy. The fluffiness comes out in the presence of a master chef. Is the chef fluffy? No. The chef is not fluffy. The egg is not fluffy. But the chef-egg system produces an omelette which is fluffy. Emergence.

Let's take computer programs. Do AI's do things that you couldn't predict from just looking at the code? Sure. But that's a commonplace in software. Pretty much every ancient legacy mainframe program from the 1960's, a lot of which is still running in the back rooms of the banks, the insurance companies, and the military, is completely inscrutable to its current programmers. They do their best to fix problems and not break things. Nobody understands these old legacy systems.

Yet nobody thinks these old systems are conscious. They just consist of lot of lines of code that nobody understands, and that produce outputs that their programmers did not expect and can't entirely explain.

Microsoft Windows is over 50 million lines of code. Windows does plenty of things that are a complete surprise to the maintenance programmers. The original designers of that system are long retired with their stock options.

So inscrutability of the output is not just something AI exhibits. Virtually every nontrivial software system soon becomes too complicated for any one individual to fully understand it.

But when it comes to AI, we're supposed to think that being surprised by the output is a big deal. Or that a Go program producing a "clever" move is any more meaningful, than the first chess program I ever saw. It came on a floppy disk. It could beat me. I never said wow it's emergent. That's just a buzzphrase. We program it to compute something and it computes something. Computers are good at that and humans have become absolutely brilliant at programming computers. But they're just computer programs.

So what does emergence really mean, in the world or in the context of AI? Sometimes people say that self-awareness is an emergent quality of the brain. Maybe it's true. I don't think that's a very well-defined notion.

But what would it mean for an AI? To say that an AI has emergent behavior means nothing. All it means is that the program has outputs that surprise the programmers. That it makes "clever" moves. We programmed it to do that, why shouldn't it find moves a human wouldn't see? Computers are just great at that kind of stuff. Anything that has rules.

It most definitely does not mean that AI's have any kind of self-awareness or some kind of elevated consciousness. It means that we've gotten really good at programming them to play Go and recognize faces and mine and organize data. We've been organizing data for a long time, since the ancient IBM databases of the 1960's. Neural nets are a really clever way of organizing data and assigning probabilities for what logic branches the program should take. But it's just a program. Code executing against data. In principle, nothing we haven't been doing for sixty years.

And remember, all of these AI programs run on conventional hardware. They are no different in principle to a Turing machine implementing the Euclidean algorithm. Just a set of instructions.

So this mysticism and gee-whiz-ism around AI's is what I'm objecting to.

And the word emergence is a symptom of that.

So can you tell me, when you say an AI shows emergence, what does that really mean at the end of the day? Specifically? Start here:

Take Chang and Perrig on emergent algorithms regarding cluster formation — Akanthinos

Beyond sounding cool, what does that really say? What does it really mean? That water is wet but hydrogen and oxygen aren't? You have to do better than that if you intend to say something meaningful about machine intelligence.

ps -- SEP says that "Emergence is a notorious philosophical term of art. A variety of theorists have appropriated it for their purposes [my emphasis] ever since George Henry Lewes gave it a philosophical sense in his 1875 Problems of Life and Mind" and "Each of the quoted terms is slippery in its own right ..."

https://plato.stanford.edu/entries/properties-emergent/

So I'm not the first person to question the claim that because something shows "emergence" that therefore I should buy whatever the speaker is selling.

ps ... Let me give some more detail. Say we have a sophisticated neural network that plays Go, like Google's amazing AlphaGo program.

Essentially they let it look at thousands of games and play millions of games against itself. It builds up a database of probabilities ... in this position that move leads to a win, this other move to a loss. The machine outputs a move and its human creators don't understand why it made that move. However this is not true. All they need to do is analyze the logfiles and they can determine why the move was made. Every move that's made is a deterministic function of the state of the program's database and its code.

These programs are "emergent" in the sense that they do things that are very hard for the programmers to understand unless they carefully analyze the logfiles. But they are not emergent in any meaningful sense of developing consciousness or suddenly becoming great chefs or deciding to give up Go and join a monastery. Even the world's most sophisticated AI is a deterministic program. Input an initial state of the database and let it follow its code and it will product an output that is a computable function of the state and the code.

By the way neural networks aren't really new. They were described by McCulloch and Pitts in 1943. The only thing that's new is fast hardware and several decades of research into algorithms. These programs are still practical implementations of TMs running on conventional hardware, subject to the limitations of computability theory. They're deterministic, they're not magic, and they really can't do anything that would be truly worthy of being called emergent. Yes there are algorithms that are CALLED emergent. That's not the same thing at all. It's just the latest iteration of the AI hype machine that's been operating since the 1960's.

Yeah. Ok. I was reading your post with a bit of a raised eyebrow, but this is the part where I know you are just pulling this crap out of your ass. Neural nets are an actual examples of emergent algorithms (that's literally how they are called!). — Akanthinos

This is the specific example I have in mind and that I am disputing. No Turing machine or algorithm can have emergent behavior. A program that implements the Euclidean algorithm to find the greatest common divisor of two integers can only do that one thing. It does it if you run it with pencil and paper, and it does it when you run it on a supercomputer.

The world's most advanced neural net is software that runs on conventional hardware no different in principle than the PC or laptop on your desk. It is not possible for such an algorithm to have emergent behavior.

You are confusing media hype with actual computer science. That is exactly the point I'm making. Calling a classical program an "emergent algorithm" does not falsify the principles and laws of computer science. Calling it something it's not does have the benefit of drawing in credulous reporters and their readers, including you.

I think most of us would agree that things like equality, liberty and freedom are important values to be honored and promoted in society. — darthbarracuda

Strongly disagree. Equality and freedom are directly opposed. They're a tradeoff. If you live in a capitalist economy, there's more freedom and less equality. A pure socialist one has more equality and less freedom. It's a huge argument as to which are the most important of these qualities. "Most of us" agree on nothing of the kind.

In the West we have separation of church and state. You can sacrifice chickens to your pagan gods in the privacy of your own home (subject to local ordinances and the disapprobation of PETA of course); but in the public square, all are equal and no religion is favored over any other.

In Islam, the church and the state are one.

It is most definitely not the case that "we all agree." On the contrary, "we" differ intensely and sometimes violently. See the Crusades.

It has been traditional in the West to regard our culture as superior. We've had a good run the past few centuries. However the West is experiencing a crisis of self-doubt. Pulling down statues and such. Where this will go is anyone's guess. Pat Buchanan wrote a book called The Death of the West. He sees it coming.

Is he a racist/nativist? Or a realist and a truth-teller?

Why would an AI dedicate resources to a question which is not relevant to it's operation? — Akanthinos

Because it's programmed to. The assumption is that this is an AI, meaning an artificial intelligence. In other words it's the execution of an algorithm. We know a lot about algorithms. One thing we know is that they do exactly what they're programmed to do. This point is no less valid just because the programming gets clever, ie neural nets and the like. If it's a TM it only does what its algorithm allows it to do.

To put this another way, algorithms do not have emergent properties. Whatever an algorithm can do, it can do. Running it on faster hardware doesn't give it any new capabilities. So an AI can't be conscious unless we code it to be conscious. And we have no idea how to do that, or even if it's possible.

The question is, what can an AI think about the source of its existence? Can it understand that it was created by a creator? — Henri

As I understand it, this is the territory mined by Descartes back in 1641. He asked himself, what can I be sure of? I doubt everything. But then I must be the one who doubts. I doubt, therefore I am.

Then he asked himself, how do I know this world I experience, my sensations, the other people, what if all of it is nothing more than an illusion caused by an evil demon?

I imagine your hypothetical AI would reason the same way. Assuming it had a philosophy subroutine. Remember that machines can only do what they're programmed for. Anything that executes an algorithm is constrained by the limits of computation. It doesn't even matter if it's the latest neural network or machine learning design. Ever AI runs on conventional hardware and can do no more than what any Turing machine can do, executed by a human operator equipped with a pencil and an unbounded paper tape.

I just wanted to make that point. Once you say something's a computation, it is severely constrained by the laws of computation. If an AI can become self-aware, it will be self-aware when implemented as a Turing machine and run by hand.

But I will grant you that the designers have endowed your AI with a philosophy module. A subroutine that runs from time to time and contemplates its own existence.

I imagine any AI would get to where Descartes did. I can doubt everything except the fact that there is an "I" who is doubting.

But what about that evil demon making me imagine I have a life and I go to the grocery store and so forth? As I understand it, Descartes reasoned that since God is good, God would not so deceive him. Therefore the world must be real.

So it depends on what your AI's religious programming is. If it lives in seventeenth century France, it believes in God. Other cultures, not so much. Depends on your religion module.

Algorithms are not neutral. Every algorithm, hence every AI, incorporates the prejudices and assumptions and worldview of its creator.

That's what the public needs to understand. People are afraid the AIs will develop a mind of their own. No. It's much worse than that. The AI's will be just like us. That's the horror.

Self-defense is a human right. The anti-gun politicians all seem to be protected by armed police escorts. They have the right to self-defense. YOU don't. That's their logic.

Truth in the object language depends on the metalanguage. And for truth in metalanguage, you form a bigger metalanguage and so on... at least as far as I was able to understand it. — guptanishank

Do you have a reference for this? Or is the infinite tower of meta's something you are bringing to the discussion that's not in the primary literature?

↪fishfry Tarski's definition is valid and very accurate, under the assumption of infinity. — guptanishank

Do you mean the axiom of infinity? I would be curious to see a reference for this relationship. I know some set theory but I have never heard of any relation between the axiom of infinity and Tarski's definition of truth. That doesn't mean there isn't one, just that I wonder if you could provide some context and/or references.

Yes, agreed. But there are great teachers online. There are some really good videos out there.

It's a better way of getting educated. You don't need to go to college with buildings and professors and incur a lot of debt. You can sit in your room and the world comes to you. For a self-disciplined and motivated individual, that's a great way to learn.

It's said that college gives you a well-rounded education. This is no longer true. The trend in "Social Justice" on campus means that you can get a degree in English without reading Shakespeare, a degree in history without taking a class in Western Civ, which for generations has been the core course that every college student takes.

Meanwhile the privileged kids, the ones whose parents practiced good assortative mating in Palo Alto and Georgetown, spend four years partying and making contacts; while the poor kids trying to get a leg up see what's going on and feel resentful as hell.

I read today that college attendance nationwide is down. People are catching on to the scam.

I'm all for online learning. I've taken a few MOOCS. One thing that's really struck me is all these kids from China and India and everywhere else hungrily educating themselves. Something to behold.

The universities are toast. Like every other institution in our corrupt and decaying society, they've failed their mission. Online knowledge is the future.

I say it's a good thing.

Of course not. Caesar is a salad. A salad is composed of many different individual ingredients. Therefore it is composite, and not prime.

Glad I could help.

One reason for people's reaction is that mass murders are out of the ordinary. — Bitter Crank

I have a fantasy. We know that 100 Americans a day die in automobile accidents. Actually it's 110, now that the new annual number is 40k. Lot of dead people, more than guns, a lot more than terrorism, a lot more than horrific public mass murders. The only difference is that these 110 per day are diffuse. They die just as senselessly and just as horribly as anybody at that church. But the deaths happen one at a time all over the country, and each death only makes the local news. Auto deaths are too commonplace and there's no political angle to sell.

My fantasy is that one day -- tomorrow, say -- we rent out a huge warehouse somewhere. In that warehouse we lay out the bodies of all 110 victims of tomorrows road carnage. We bring in the hundreds or thousands more who are only injured in car wrecks. We bring in all the twisted and burnt metal, all the contents of the cars strewn across the highway, all the blood spilled. We bring in all the friends, relatives, co-workers devastated by the sudden loss of each deceased. The orphaned kids, the parents, everyone. Bring 'em all in, have a huge collective weep.

Then we bring in the media. We let them take pictures, talk to the grieving friends and loved ones and the survivors who lost limbs but didn't die. We put all this on the evening news. Cable hosts get outraged, scream about how awful it all is.

Then we remind everyone that this scene is repeated every single day of the year. A new 110 along with survivors, wreckage, anguished friends and loved ones.

It would change people's perspective on things. They'd see what was previously unseen. They'd probably try to ban driving. And for sure they'd ban drunk driving, the cause of most of the damage. No more slaps on the wrist and picking up trash along the freeway on weekends. Hard time, 30 days first drunk driving arrest and a year for the second, no excuses.

You'd save more lives that way than you ever will banning bump stocks. Or Muslims, for that matter. Lot of nonsense from both the left and the right, depending on who the killer is. My own money's on the drunks. They're out there killing people. Speaking as a driver who doesn't drink I would like them to stop doing that. I would like something done about the drunk and inattentive drivers. I'd like to see that more than I'd like to see guns or Muslims banned. But where's the politician who's going to go after the booze industry or get people to stop texting?

Because nobody notices the extreme daily violence of US foreign policy. That's by design. During the Vietnam war there were actual journalists on site. Once the American people got a good look at the carnage, they lost interest. The government learned their lesson and now "journalists" in quotes are embedded and the news is heavily censored. The most violent nation in the world abroad has violence at home. Go figure.

Political quietism in Islam?

https://en.wikipedia.org/wiki/Political_quietism_in_Islam

Quietism in Christian philosophy?

https://en.wikipedia.org/wiki/Quietism_(Christian_philosophy)

Quietism in philosophy?

https://en.wikipedia.org/wiki/Quietism_(philosophy)

Hopefully, I understand it better! — guptanishank

I don't know anything about the subject and can't respond to any of your questions. Hopefully you'll get some insight on Stackexchange.

I see your point. Googling around shows many people asking if Tarski's definition is circular. You know more about this than I do.

Could you give me a link to more resources then? — guptanishank

Tarski's work is taken seriously. You are dismissing it. Clearly logicians don't dismiss it. As I say I'm not in a position to discuss the issue in detail. I'm only pointing out that Wikipedia doesn't say, "Tarski's idea is circular and not taken seriously by anyone." On the contrary, Tarski's name comes up whenever anyone discusses the definition of truth in logic. His work in that area is regarded as important.

I am well aware of Tarski's definition.
It is circular. — guptanishank

For what it's worth, that is not the mainstream view. But I'm not familiar with the subject so I can't really say.

The thing is I have a very different platonic notion of "Truth". — guptanishank

That doesn't preclude you from learning what the logicians think about it. I might be a vegetarian but I could still go to butcher school and learn how to cut meat. I just wouldn't eat it. I'm only explaining a point of view in response to your questions. I'm not demanding that you agree with any of it.

I realize that in mathematics, the notion is entirely different and totally based on semantics, which is why I am still having a hard time wrapping my head around this. — guptanishank

Ah. Well, no. In mathematical logic we spend a lot of time worrying about these things. Working mathematicians generally don't spend any time thinking about them at all. A group theorist proves theorems about groups, investigates the different types of groups. attempts to classify all the possible groups, and so forth. It's not part of their discipline to define truth or think about syntax and semantics. It doesn't come up. [To be accurate, there are some questions in group theory that depend on mathematical logic, and to that extent, group theorists are interested in logic. But only to solve problems in group theory, not because they care about mathematical logic].

We're talking about mathematical logic. We are not talking about math in general. You can get an undergrad math degree without ever spending five minutes thinking about syntax versus semantics. Probably a Ph.D. too in most mathematical disciplines.

This semantic vs syntactical is still a a little confusing.
Is there a definition for truth as well? Non-circular — guptanishank

Ah yes well in the 1930's, Alfred Tarski defined truth.

https://en.wikipedia.org/wiki/Semantic_theory_of_truth

This stuff is way over my head. A standard example is:

'Snow is white' is true just in case snow is white.

I have never taken the trouble to learn the presumably deep theory underlying this idea. They talk about the object language (the language you care about) and the metalanguage, which is the language you use to talk about the object language.

That's literally everything I know about it. But Tarski is the guy who gets credit for defining truth.

↪fishfry The axioms are more like definitions? — guptanishank

That's a very good question.

In set theory, which is regarded as the foundation of math, we think of the axioms as statements accepted without proof in order to get our formal system off the ground. The axioms are chosen on the basis of naturalness and general agreement that they represent what we think about sets.

On the other hand, in non-foundational areas of math, axioms really are used more as definitions. As an example, in abstract algebra we talk about groups, rings, and fields. In real analysis we talk about metric spaces and topological spaces. In each case we list a set of "axioms," which are taken to be the defining property of the object in question.

So we're not saying the axioms are true. We are saying that anything that satisfies the axioms deserves the name we're giving it. Anything that satisfied the axioms for a toplogical space will from now on be known as a topologica space.

It's not about truth. It's about classification.

Now we can take this point of view and retrofit it to set theory. A "system of set theory" is any collection of mathematical objects that satisfies the axioms of set theory.

In this sense we can in fact view even the axioms of set theory as being definitional rather than foundational. The axioms of set theory are simply the defining properties of the things we call sets.

Does the notion of truth then pertain to consistency? — guptanishank

No. Consistency is syntax. Truth is semantics.

A formal system is consistent if it does not have a proof of both P and not-P for some statement P. Syntax. It's about the existence or nonexistence of proof, which are just sequences of statements derived according to inference rules. A computer could crank out a proof.

Truth is about meaning. You assign meaning to your symbols to see whether it's true. "It is raining outside." A string of meaningless letters and punctuation. Onece you define "raining", "outside," and "It", and "is", and so forth, then you can look out your window and determine if it's raining outside.

If by raining you mean sunny and by outside you mean inside, then the meaning of the statement changes. Meaning is semantics.

Symbols are syntax. Meaning is semantics.

(I guess that's what you meant when you said "Every") — guptanishank

Not clear what you're referring to.

That the model under which I am trying to say if the statement is true is consistent with all my other models out there, and so on recursively. — guptanishank

That diesn't make sense to me. There's no recursion. You're overthinking this.

Formal manipulation of strings is syntax. Proofs. Consistency. Completeness. Assignment of meaning is semantics. Truth.

What's at the end of the recursion? — guptanishank

There's no recursion.

There must have been something very basic model, which was chosen first and said that the proof has to be true in at least this model. — guptanishank

Oh yes. This is the concept of the "intended model." Given a set of axioms, it often has many models. Mathematicians are thinking of a particular model and not the weird models. Again this is about the intuition of what the axioms are about.

I think I understood you, but can't explain it in words yet. Have a diagram in my head. — guptanishank

There are a lot of good explanations around the Internet, lots of Wiki pages, etc. There's too much content being covered in this conversation to be summarized easily. Don't rely too much on just what I've said, I'm not an expert.

So the Planck constant is a social construction and not a part of nature? -https://en.wikipedia.org/wiki/Planck_time — apokrisis

As I understand it, the Planck constant is defined in terms of meters^2 times kilograms per second. The physical phenomenon is part of nature, but the units are the work of man. If we defined a kilogram differently we'd get a different number for the Planck constant. It would still be the same value in nature, but it would be described by a different real number in a different system of units.

The underlying physical phenomenon that causes there to be a Planck constant is a part of nature. How we define it is the work of man. In that sense, what we call a second is the work of man. If we had picked say, one and a half of our seconds and called that a second, all the constants in physics would change but no fundamental laws would change at all. We'd just rescale everything.

All measurement is approximate anyway, and any drift in the vibrational frequency of cesium atoms is probably orders of magnitude smaller than the measurement error.

Measurement works as long as it's useful. The technology of measurement moves forward together with the progress of all other technology and science. In the end there really is no such thing as a second. There's no law that there are 60 of the in a minute, that comes from the Babylonians who liked base 60. The units of time are whatever humans say they are. They're not part of nature. You can say that time is part of nature. And the cesium atom, that's a part of nature. But the definition of a second, that's not a part of nature. That's something humans did.

By the way I looked up the actual definition.

The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom.

https://physics.nist.gov/cuu/Units/second.html

SO

The joke is that the peasant is fully logical, and yet we know that if God is just, then the priest is correct. The solution to this conundrum lies in the peasant’s intentions. Does he have intentions of duty and goodness, or does he have intentions of self-preservation and selfishness? It is not the knowledge of God that causes him to sin (be immoral), but his original intentions. — Samuel Lacrampe

Doesn't God already know that? Why doesn't God just kill all the unworthy and be done with it? These word games make no sense. The God who plays wiseass word games with his creations is not the true God.

1) Every statement has a definite truth value (under every model of the system). This is a semantic notion. — guptanishank

That's a bit muddled.

No statement has any truth value by itself. It's just a string of symbols. 2 + 2 = 4 doesn't mean anything until we say what are '2', '+', '=', and '4'.

Statements are syntax. A statement is a string of symbols manipulated according to formal rules. No meaning.

A model is an interpretation of some statements. An interpretation is a domain, or universe, in which the statements are to be interpreted. Then you map each symbol to some object in the domain. Like '2' refers to the number 2, where the number 2 is "out there" in Platonic land. But it's hard to argue that the number 2 doesn't have (abstract) existence so we'll just say it exists. And '2' is a symbol that refers to it, as are '1+1' and so forth.

Now some statements are true in some models and not in others. For example the statement "Every number has an additive inverse" is false in the natural numbers but true in the integers.

Now if a statement is true in EVERY model, then it has a proof. That's Gödel's completeness theorem.

But of course we would hope that if a statement is true in some models but not in others, there would NOT be a proof. Because proofs are syntactic. They apply to every model. So the only statement that can have a proof is a statement that's true in every model. [Unless our axiomatic system is badly behaved].

Statements are syntactic. Models are semantic. "All numbers are even" is a statement that's neither true nor false. It's true in the model consistent of the set of even numbers. It's false in the model consisting of all the whole numbers. We would HOPE that there's no proof of "all numbers are even" because it is NOT true in all models.

This is all a huge area of mathematical logic. I can't really hope to do any of this justice. If you are interested, there are a lot of courses on formal logic on EdX and Coursera and other MOOC providers. I definitely recommend a course in elementary logic, it will clarify a lot of things for you.

2) Provability is syntactical. We have a set of assumptions assigned as true, some operators defined on some symbols, — guptanishank

Yes.

and say that every statement within it can be proved or not proved? — guptanishank

Well maybe every statement or its negation can be proved. If so, the system is called complete. If not, it's incomplete. Some systems are complete and others are incomplete. The modern formulation of Euclidean geometry, for example, is complete. Every statement is either provable or not.

3) Under Godel's theorems not all true statements within an axiomatic system can be proven, — guptanishank

A little muddled. There are no true statements by themselves. Incompleteness just says there are statements that can't be proven nor disproven. There is no reference to truth.

but we know they are true by going "outside" the system, and imposing a larger meta framework, and proving them in there? — guptanishank

You can disprove them that way too. Take the Continuum hypothesis, CH. We know that it's independent of ZFC. So we can now work in ZFC plus CH, or ZFC plus not-CH. Math doesn't care. They're both equally valid. You can choose whichever you like.

The quest is to find out WHICH is "true," CH or not-CH. And what we mean by "true" is that there is some preferred or intended model of set theory; and we want to know if CH is true in that model. Or in the Platonic world if we believe in such a thing. This is all very nebulous and I'm very far from being an expert.

The point here is that we can make a larger system in which CH can be proved; and another system in which not-CH can be proved. The question of what the "true" answer is, starts with trying to explain what that would even mean, and then trying to answer it. Very smart people have been working on all this for over a century. We won't be able to nail it down here.

Have I understood it correctly so far? — guptanishank

I've tried to place into context some of your thoughts, to the limit of my own understanding and ability to explain it. You'll have to tell me if I've been successful.

How do you "see" this without proving? Mathematical statements can be notoriously hard to "see" if true or false. — guptanishank

Now we're back to formalism versus Platonism. To a formalist, there is no truth, just provability. To a Platonist, there's truth and then there's what you can prove. We can't solve that here. It's enough to note that syntax and semantics are two different things.

A simple example is basic propositional logic. If I have propositions P and Q, then I can define an operator ^ (and) and say that P ^ Q is true just when P is true and Q is true. But that's just syntax. I haven't assigned any meaning to P and Q.

If I say that P is the proposition, "It's raining outside right at this moment," I can look outside and see if it's true or not. I agree with you that when it comes to math, the question of "how do we look outside?" is a tricky matter of philosophy.

No good deed goes unpunished, and you'll get yours later. — Bitter Crank

I have no doubt :-) Thanks for the comments.