Misunderstanding basic math concepts, help please?
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 gmalivuk
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Re: Misunderstanding basic math concepts, help please?
Taste results from a chemical reaction, and the basic chemical properties of sodium chloride are not the simple sum of the chemical properties of sodium and the chemical properties of chlorine. Ionization changes chemistry, as does combining with other atoms. I don't know a lot about chemistry, but I know that much.
Like, water isn't a gas even though hydrogen and oxygen are gases. Are you seriously arguing that basic physics is unable to explain that fact?
Like, water isn't a gas even though hydrogen and oxygen are gases. Are you seriously arguing that basic physics is unable to explain that fact?
 Xanthir
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Re: Misunderstanding basic math concepts, help please?
Uh, the fact that taste is based on chemical reactions and molecular shapes, not directly experiencing the individual atoms? Like, we can explain mathematically why kids liked to eat lead paint  lead oxide has a particular triangular shape that mimics the shape of parts of the sucrose molecule such that it tricks those taste buds.
Or what gmal said. (Preninja'd.)
Or what gmal said. (Preninja'd.)
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 Cleverbeans
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Re: Misunderstanding basic math concepts, help please?
gmalivuk wrote:Like, water isn't a gas even though hydrogen and oxygen are gases. Are you seriously arguing that basic physics is unable to explain that fact?
Emergent properties are a 2000 year old philosophical problem and yes, I'm arguing that they exist and reductionism fails. In the example I gave taste is the emergent property I'm talking about. Are you really arguing that you can predict the taste of something with physics?
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Re: Misunderstanding basic math concepts, help please?
Cleverbeans wrote:Lets start a short list of things you can't describe mathematically to get your started and just to be pedantic I'm going to make them all terrestrial.
Love.
Aesthetics.
Morality.
The placebo effect.
Pop culture.
Musical tastes.
Yo mama jokes.
Vellichor.
Ambition.
Schizophrenia.
Acting.
Justin Beiber.
Pedagogy.
Creativity.
Child abuse.
Petrichor.
Twerking.
Silliness.
Boredom.
Religion.
Need more?
Not sure what's the point of this list is, since I've already stated myself that subjective experiences are one of the things that at least seem to be beyond math's power.
You can write a list of a million other things like that, while I can write a list of million things that math can describe. But what's the point? Since none of us can actually list more than a tiny fraction of everything there is, niether of us could get very far in this way.
Cleverbeans wrote:Emergent properties are a 2000 year old philosophical problem and yes, I'm arguing that they exist and reductionism fails. In the example I gave taste is the emergent property I'm talking about. Are you really arguing that you can predict the taste of something with physics?
Theoretically, yes.
After all, we can definitely predict how salt will look (cubic white crystals) from basic physics.
And anything you could argue about taste, can also be argued about looks:
1. "looks" is a subjective preception.
2. Sodium and chlorine both look very different than salt.
Are you seriously arguing that physics can't explain why salt is white?
gmalivuk wrote:The composite number 21 plus the composite number 10 sum to the prime number 31, while the prime 5 plus the prime 7 is the composite 12. That doesn't mean the properties of numbers are some kind of mystical thing that can't be explained by "reductionist arguments", it just means compositeness and primality aren't the sorts of things preserved by addition.
An even better example would use multiplication rather than addition.
After all, primes are the "atoms" of number theory. Any number can be broken down into primes in a unique way.
We can even model emergent properties in this way:
The numbers 3, 3607 and 3803 don't seem too interesting on their own. But if we multiply them (using '3' twice) we get 123456789 which "tastes" completely different
 gmalivuk
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Re: Misunderstanding basic math concepts, help please?
You've moved the goalposts.Cleverbeans wrote:gmalivuk wrote:Like, water isn't a gas even though hydrogen and oxygen are gases. Are you seriously arguing that basic physics is unable to explain that fact?
Emergent properties are a 2000 year old philosophical problem and yes, I'm arguing that they exist and reductionism fails. In the example I gave taste is the emergent property I'm talking about. Are you really arguing that you can predict the taste of something with physics?
I said I could justify why salt doesn't taste like the taste of sodium mixed with the taste of chlorine, not that I could personally predict the subjective experience of something's taste from first principles. (If all of your examples hinge on a subjective experience somewhere, then the only thing you're really arguing is that science doesn't yet have a full explanation of how consciousness works, and you're relying on that single gap for your entire list of "inexplicable" emergent phenomena.)
However, I definitely can tell you things *about* the taste experience of salt, as compared with the taste experience of sodium or chloride. For example, small amounts of salt won't come with a pain sensation, even though both sodium and chlorine would, because salt isn't reactive enough to physically damage the surface of the tongue, and that's one of the things that leads to the sensation of pain when tasting something.
One more knowledgeable than me could go further and say things about the nerve impulses sent by the tongue and taste buds, based on knowledge of chemistry and the physical structure of the relevant cells.That wouldn't get you all the way to the subjective experience of "what salt tastes like", but it would for damn sure explain why that subjective experience is so different from the experience of "what pure sodium tastes like".
Which, as mentioned, was your original "challange" before you moved the goalposts.
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Re: Misunderstanding basic math concepts, help please?
Cleverbeans wrote:gmalivuk wrote:Like, water isn't a gas even though hydrogen and oxygen are gases. Are you seriously arguing that basic physics is unable to explain that fact?
Emergent properties are a 2000 year old philosophical problem and yes, I'm arguing that they exist and reductionism fails. In the example I gave taste is the emergent property I'm talking about. Are you really arguing that you can predict the taste of something with physics?
Yes, we can. I literally just gave an example of that. Flavor scientists do that right now. Flavor isn't a magical mysterious force separated from physical reality.
(Tho actually it would be kinda cool if one was a dualist, but only for flavor. "Souls? No, that's stupid. But the taste of a perfect cappucino, an ideal risotto, a sublime ice cream sundae... That transcends petty physical reality.")
Edit: goddammit gmal, stop responding in between the time I open the topic and when i press "submit"
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 gmalivuk
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Re: Misunderstanding basic math concepts, help please?
Also, we all know how Wikipedia works. Posting a link to the Emergence article doesn't actually accomplish anything. And reminding us that it's something people noticed a long time ago sure as hell doesn't get you to your unscientific conclusion that reductionism fails.
(Actually, if you feel the need to define something, it should probably be what you mean by "reductionism". Because I'm pretty certain no one else here takes it to mean you can just apply the laws of quantum mechanics a bunch of times and then directly read off a subjective description of the flavor of salt. But that seems to be the straw version you're arguing against.)
(Actually, if you feel the need to define something, it should probably be what you mean by "reductionism". Because I'm pretty certain no one else here takes it to mean you can just apply the laws of quantum mechanics a bunch of times and then directly read off a subjective description of the flavor of salt. But that seems to be the straw version you're arguing against.)
Re: Misunderstanding basic math concepts, help please?
That wouldn't get you all the way to the subjective experience of "what salt tastes like", but it would for damn sure explain why that subjective experience is so different from the experience of "what pure sodium tastes like".
Getting "all the way" to the subjective experience may very well be impossible without additional assumptions. It require a shift of perspective that our current framework of physical concepts cannot handle.
But even that doesn't say anything about the power (or lack of) mathematics. It simply says that right now we don't have a clue about how "the physics of the subjective" might work (although we have a tiny clue in the form of the wave function collapse in quantum mechanics)
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Re: Misunderstanding basic math concepts, help please?
...what? No, "subjective experience" has nothing to do with waveform collapse. Like, at all, that wheelhouse is miles away. Subjective experience is just something that pops out of our brain patterns. There's zero need to invoke quantum mechanics at all, and most scientists afaik believe the brain is too hot/wet (technical terms) to depend on largescale quantummechanical effects at all.
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Re: Misunderstanding basic math concepts, help please?
I didn't say that subjective experience necessarily has anything to do with waveform collapse.
I simply said that subjective experience is outside the realm of description of conventional physics. Even if we had a "map" that tells us exactly what brain state corresponds to what subjective experience, it still won't explain why THIS brain state corresponds to THAT experience.
And quantum mechanics gives us a tiny peak at a possible alternative approach. The "collapsebymeasurement" equation doesn't speak of an interaction between particles. It speaks about how the universe changes in response to A REQUEST OF INFORMATION.
What this really means in the physical sense  nobody knows. Whether this has anything to do with the "subjective physics" problem  ditto. But QM, at least, is speaking in a strong enough language to potentially deal with these problems.
I simply said that subjective experience is outside the realm of description of conventional physics. Even if we had a "map" that tells us exactly what brain state corresponds to what subjective experience, it still won't explain why THIS brain state corresponds to THAT experience.
And quantum mechanics gives us a tiny peak at a possible alternative approach. The "collapsebymeasurement" equation doesn't speak of an interaction between particles. It speaks about how the universe changes in response to A REQUEST OF INFORMATION.
What this really means in the physical sense  nobody knows. Whether this has anything to do with the "subjective physics" problem  ditto. But QM, at least, is speaking in a strong enough language to potentially deal with these problems.
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Re: Misunderstanding basic math concepts, help please?
That's not how quantum mechanics works, qualia aren't real, and pzombies beg the question.
Now let's get back to discussing math.
Now let's get back to discussing math.
Re: Misunderstanding basic math concepts, help please?
I have a degree in physics and I aced QM. Qualia are real  subjectively. And I don't believe pzombies make sense.
And I don't think that rediculing an opinion with a kneejerk reaction and immediately saying after that "enough of that, let's talk about something else" in official red letters is fair.
(I agree, btw, that we got sidetracked. But really, did you have to be so disrespectful while doing that?)
And I don't think that rediculing an opinion with a kneejerk reaction and immediately saying after that "enough of that, let's talk about something else" in official red letters is fair.
(I agree, btw, that we got sidetracked. But really, did you have to be so disrespectful while doing that?)
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Re: Misunderstanding basic math concepts, help please?
You're welcome to start another thread about how the equations of QM somehow encode "requests" for information.
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Re: Misunderstanding basic math concepts, help please?
PsiCubed wrote:(I agree, btw, that we got sidetracked. But really, did you have to be so disrespectful while doing that?)
It's important to be disrespectful of nonsense, because it is easy for nonsense to get the wrong idea about its place in the dialog.
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Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
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Re: Misunderstanding basic math concepts, help please?
I googled "does salt taste like sodium and chlorine together" and literally the first result is a Stack Exchange question whose answer effectively says that yes, salt tastes like sodium cations and chlorine anions together because the salty taste is triggered by the sodium cations once the ions separate in the aqueous solution that is your saliva.
So reductionism wins again?
So reductionism wins again?
(∫p^{2})(∫q^{2}) ≥ (∫pq)^{2}
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 Cleverbeans
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Re: Misunderstanding basic math concepts, help please?
gmalivuk wrote:You've moved the goalposts.
I very clearly said what I meant the first time and I repeated myself word for word. Frankly between your condescending remarks and arbitrary dismissal of my arguments I don't see any more point in addressing your comments.
Xanthir wrote:Yes, we can. I literally just gave an example of that. Flavor scientists do that right now. Flavor isn't a magical mysterious force separated from physical reality.
I do concede that salt was a poor example. Between it's trivial structure and the fact it ionizes in the mouth so we should have a reasonable expectation that it tastes like it's components. It's also one of the most studied flavors since like sugar substitutes are often sought after.
As far as I know they can't predict the taste of an arbitrary molecule but I sent an email to the Society of Flavor Chemists just to verify this. I believe powers are limited to examining known flavors and building molecules that interact in a similar way. If that's the case starting with the answer you're looking for is a significant advantage. At first glance bitterness seems to be particularly sensitive to small changes in molecular structure. I'll let you know what I find out.
More so taste remains highly subjective. Our experience of taste emerges in a way that isn't fully understood even though we have a reasonably strong understanding of it's chemical and neurological underpinnings. Parageusia would be an extreme example of this.
PsiCubed wrote:Not sure what's the point of this list is, since I've already stated myself that subjective experiences are one of the things that at least seem to be beyond math's power.
To get you to agree to his. Subjective experiences are a large part of the universe, math can't explain much of it. Your claim that most of the universe can be explains mathematically breaks really quickly once we toss them into the mix. Your myopic view of the universe ignored this. In any event I don't think this line of reasoning goes any further since we both agree on this.
Theoretically, yes.
Speculative. Ignorance is the natural state of human beings. If you're saying we have access to a certain type of knowledge the burden of proof is yours.
After all, we can definitely predict how salt will look (cubic white crystals) from basic physics.
The ability to predict some physical properties doesn't imply the ability to predict others without justification. This is very handwavy.
An even better example would use multiplication rather than addition.
Using mathematical metaphors to justify why the universe is like math is circular. You'll need to find a justification outside of mathematics as to why the universe conforms to your intuition here.
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Re: Misunderstanding basic math concepts, help please?
The number examples weren't supposed to singlehandedly justify why math explains everything, they were supposed to illustrate why your naive salt example was so logically weak. You suggested that the properties of a combination of things should be expected to be the combination of the properties of each thing individually. Our point was that this isn't even true in the simplest, most transparent of cases (mathematical ones), so there's no reason to expect it to hold true in more complex cases. Emergent phenomena exist, but you picked a poor illustrative example.
Also, the bigger part of the reason we can't predict flavors from first principles is that the mathematical modelling of the chemical reactions that take place on the surface of the tongue gets devilishly complicated. The subjective side could often be tackled by analogy ("this thing will taste kind of like that thing with a hint of this other thing") if we were able to get to the point of knowing how the taste buds would fire their nerve signals.
But lots of things are impractically difficult to calculate without it meaning there's a failure of reductionism. We know all about what it means for one number to be a factor of another, but that doesn't make it any easier to factor large semiprimes in practice.
(Again, I'm arguing by analogy. Your example is a bad one for the same reason "we can't factor large semiprimes" is a bad argument against reductionism in mathematics.)
Also, the bigger part of the reason we can't predict flavors from first principles is that the mathematical modelling of the chemical reactions that take place on the surface of the tongue gets devilishly complicated. The subjective side could often be tackled by analogy ("this thing will taste kind of like that thing with a hint of this other thing") if we were able to get to the point of knowing how the taste buds would fire their nerve signals.
But lots of things are impractically difficult to calculate without it meaning there's a failure of reductionism. We know all about what it means for one number to be a factor of another, but that doesn't make it any easier to factor large semiprimes in practice.
(Again, I'm arguing by analogy. Your example is a bad one for the same reason "we can't factor large semiprimes" is a bad argument against reductionism in mathematics.)
Re: Misunderstanding basic math concepts, help please?
Man things have really gotten out of control here. Let's try to get back onto the original topic of the fitness of formal logic as a foundation for mathematics.
Here's what happened as I see it (and using my definitions of terms).
It appeared to me that Treatid thought math or mathematicians claim that mathematics can access universal truth. It also appeared to me that a lot of Treatid's argument against mathematics hinged on this belief of his. I agree with him that if that were a claim made by mathematicians he would have a lot of ammo against axiomatic mathematics. At that point I was trying to make the argument that no mathematicians think mathematics can access universal truth. I said he was basing his whole argument against a platonistic strawman that didn't actually exist.
I guess that's where things went awry. PsiCubed took a bit of offense and pointed out that actually a lot of mathematicians are platonists and as the discussion went on also pointed out that these mathematicians think math can access absolute truth (for some definition of absolute truth).
I think this is a critical part in our discussion with Treatid. Usually the nonTreatid posters present a unified front against Treatid's attacks against axiomatic mathematics and try to explain to him what he's missing about how mathematicians actually think about math, but in this case. As the past few PAGES of this thread have shown, we are not unified. Basically it seems like all parties have different philosophical viewpoints when it comes down to these questions about absolute truth or whatever. I think it's an interesting point because I do think that Treatid's unwillingness to accept axiomatic mathematics as it is practiced comes from exactly this same point which has drawn contention between the rest of us.
Anyways, back to what happened. After we opened the can of worms on that platonism/formalism debate there was a bit of discussion about the meaning of absolute/universal truth, absolute knowledge and universal languages. There were attempts to define these terms as well as discussions about how they might relate to mathematics. There were also general discussions about the power and effectiveness of mathematics which drifted into physics topics and then meandered out into stuff about reductionism and subjectivity.
Maybe I can't speak since I was the one who opened the can of worms on platonism, but I'm going to hazard that at some point between absolute truth and quantum subjectivity we got off topic.
So in any case I propose we get back to the formal logic and foundations of mathematics before the thread gets locked.
I STILL think it's important we agree on definitions of absolute truth and come to an agreement on how they relate to axiomatic mathematics. If we can't come to an agreement on that then again, I think Treatid is justified in being distrustful of axiomatic mathematics.
Honestly I think Treatid is just a formalist and would be satisfied with that interpretation of mathematics if he came to understand it.
But maybe not if he believes that humans can access absolute truth (or absolute knowledge).
Absolute/universal truth: a fact about the world which is incontrovertibly true regardless of perspective or interpretation.
Absolute knowledge: Complete epistemic certainty. Knowledge upon which NO DOUBT can be cast.
Universal language: Still not sure what everyone means by this. PsiCubed gave the following definition:
"Universal language" means that if different cultures develop it independently, they will all be speaking the exact same language and be able to understand one another.
That certainly doesn't seem like a very universally fundamental thing. It's just a statement about creatures that are able to come up with languages. PsiCubed claimed that 2+2 = 4 means the same thing on other planets but we don't know that. I'm fine with this definition of universal language, but I again just don't think it's impressive enough to warrant the title 'universe'. It's sort of like the Miss Universe pageant. The title sounds like the winner is meant to be the most beautiful woman in the universe, but the only contestants are from Earth. Clearly the "universe" in the title is just a marketing ploy. I feel the same way about may of the usages of "universal language" and "universal truth" appearing in this thread.
And before you go on to say that I'm setting to high of a bar for my definitions, I'll remind you that I am INTENTIONALLY setting a high bar for my definitions because I think they are the ones that make the most sense and can drive the discussion forward. If you want to talk about a mathematical truth call it a mathematical truth, not a universal one.
Also, apologies for basically repeating the same thing in every one of my posts. The reason I'm doing it is because I still feel like this place I keep coming back to in my posts is where we left off on our actual (ontopic) discussion regarding formal logic and it's place as the foundation of mathematics.
edit: Oh and another important point. We identified four types of mathematical questions:
1) Questions about the truthhood of certain mathematical sentences which can be proven from the axioms of mathematics. Questions like "is 85 a prime number?"
2) Questions about the truthhood of certain mathematical sentences which CANNOT be proven from the axioms of mathematics and which are independent of those axioms. This means that that mathematical sentence could be taken to be either true or false and neither would lead to a contradiction. The vocab word here is "independent". An example is the continuum hypothesis.
3) Questions about the truthhood of certain mathematical sentences which CANNOT be proven from the axioms of mathematics but which CAN be proven using metamathematics. The Godel sentence is the obvious example here but maybe Gmalivuk's questions about whether certain real numbers are normal also falls in this category.
4) Questions which are not mathematical questions at all. Things like "is Euler's formula beautiful." And don't give me some bullshit about mathematical definitions of beauty because a) you're just being obnoxious and you know it and b) I'll just come up with a more unrelated example and you know that too.
Here's what happened as I see it (and using my definitions of terms).
It appeared to me that Treatid thought math or mathematicians claim that mathematics can access universal truth. It also appeared to me that a lot of Treatid's argument against mathematics hinged on this belief of his. I agree with him that if that were a claim made by mathematicians he would have a lot of ammo against axiomatic mathematics. At that point I was trying to make the argument that no mathematicians think mathematics can access universal truth. I said he was basing his whole argument against a platonistic strawman that didn't actually exist.
I guess that's where things went awry. PsiCubed took a bit of offense and pointed out that actually a lot of mathematicians are platonists and as the discussion went on also pointed out that these mathematicians think math can access absolute truth (for some definition of absolute truth).
I think this is a critical part in our discussion with Treatid. Usually the nonTreatid posters present a unified front against Treatid's attacks against axiomatic mathematics and try to explain to him what he's missing about how mathematicians actually think about math, but in this case. As the past few PAGES of this thread have shown, we are not unified. Basically it seems like all parties have different philosophical viewpoints when it comes down to these questions about absolute truth or whatever. I think it's an interesting point because I do think that Treatid's unwillingness to accept axiomatic mathematics as it is practiced comes from exactly this same point which has drawn contention between the rest of us.
Anyways, back to what happened. After we opened the can of worms on that platonism/formalism debate there was a bit of discussion about the meaning of absolute/universal truth, absolute knowledge and universal languages. There were attempts to define these terms as well as discussions about how they might relate to mathematics. There were also general discussions about the power and effectiveness of mathematics which drifted into physics topics and then meandered out into stuff about reductionism and subjectivity.
Maybe I can't speak since I was the one who opened the can of worms on platonism, but I'm going to hazard that at some point between absolute truth and quantum subjectivity we got off topic.
So in any case I propose we get back to the formal logic and foundations of mathematics before the thread gets locked.
I STILL think it's important we agree on definitions of absolute truth and come to an agreement on how they relate to axiomatic mathematics. If we can't come to an agreement on that then again, I think Treatid is justified in being distrustful of axiomatic mathematics.
Honestly I think Treatid is just a formalist and would be satisfied with that interpretation of mathematics if he came to understand it.
But maybe not if he believes that humans can access absolute truth (or absolute knowledge).
Absolute/universal truth: a fact about the world which is incontrovertibly true regardless of perspective or interpretation.
Absolute knowledge: Complete epistemic certainty. Knowledge upon which NO DOUBT can be cast.
Universal language: Still not sure what everyone means by this. PsiCubed gave the following definition:
"Universal language" means that if different cultures develop it independently, they will all be speaking the exact same language and be able to understand one another.
That certainly doesn't seem like a very universally fundamental thing. It's just a statement about creatures that are able to come up with languages. PsiCubed claimed that 2+2 = 4 means the same thing on other planets but we don't know that. I'm fine with this definition of universal language, but I again just don't think it's impressive enough to warrant the title 'universe'. It's sort of like the Miss Universe pageant. The title sounds like the winner is meant to be the most beautiful woman in the universe, but the only contestants are from Earth. Clearly the "universe" in the title is just a marketing ploy. I feel the same way about may of the usages of "universal language" and "universal truth" appearing in this thread.
And before you go on to say that I'm setting to high of a bar for my definitions, I'll remind you that I am INTENTIONALLY setting a high bar for my definitions because I think they are the ones that make the most sense and can drive the discussion forward. If you want to talk about a mathematical truth call it a mathematical truth, not a universal one.
Also, apologies for basically repeating the same thing in every one of my posts. The reason I'm doing it is because I still feel like this place I keep coming back to in my posts is where we left off on our actual (ontopic) discussion regarding formal logic and it's place as the foundation of mathematics.
edit: Oh and another important point. We identified four types of mathematical questions:
1) Questions about the truthhood of certain mathematical sentences which can be proven from the axioms of mathematics. Questions like "is 85 a prime number?"
2) Questions about the truthhood of certain mathematical sentences which CANNOT be proven from the axioms of mathematics and which are independent of those axioms. This means that that mathematical sentence could be taken to be either true or false and neither would lead to a contradiction. The vocab word here is "independent". An example is the continuum hypothesis.
3) Questions about the truthhood of certain mathematical sentences which CANNOT be proven from the axioms of mathematics but which CAN be proven using metamathematics. The Godel sentence is the obvious example here but maybe Gmalivuk's questions about whether certain real numbers are normal also falls in this category.
4) Questions which are not mathematical questions at all. Things like "is Euler's formula beautiful." And don't give me some bullshit about mathematical definitions of beauty because a) you're just being obnoxious and you know it and b) I'll just come up with a more unrelated example and you know that too.
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Re: Misunderstanding basic math concepts, help please?
I suspect some questions about uncomputable numbers can't be proven metamathematically, either, as they would imply solutions to the halting problem. Perhaps call those questions type 3a and leave undetermined whether 3a=3.
I think Cleverbeans was trying to make a list of things about which "why" and "how" questions would be type 4, though I'd argue the list is rather cheaty since most everything on it includes subjective value judgments and it's already been acknowledged that questions like that aren't mathematical. (However, while the question, "Is this beautiful?" isn't mathematical, questions like, "What happens in the brain when we judge things to be beautiful?" and, "What is it about a formula that causes people to regard it as beautiful?" may very well be addressable by science and thus by mathematics.)
Cleverbeans thought it was clever to include only terrestrial things, but it would be cleverer to include things that don't involve the subjective element that's already been accepted as nonmathematical.
I think Cleverbeans was trying to make a list of things about which "why" and "how" questions would be type 4, though I'd argue the list is rather cheaty since most everything on it includes subjective value judgments and it's already been acknowledged that questions like that aren't mathematical. (However, while the question, "Is this beautiful?" isn't mathematical, questions like, "What happens in the brain when we judge things to be beautiful?" and, "What is it about a formula that causes people to regard it as beautiful?" may very well be addressable by science and thus by mathematics.)
Cleverbeans thought it was clever to include only terrestrial things, but it would be cleverer to include things that don't involve the subjective element that's already been accepted as nonmathematical.
Re: Misunderstanding basic math concepts, help please?
Twistar wrote:Man things have really gotten out of control here.
Yeah. Yet only my post was thrown out to another thread as "off topic".
I thought that on XKCD we could have a fair and balanced discussion, but I guess I was mistaken. I'm outta here.
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Re: Misunderstanding basic math concepts, help please?
Your post was specifically in response to my comment that you were welcome to start a new thread about it. What did you expect would happen?PsiCubed wrote:Twistar wrote:Man things have really gotten out of control here.
Yeah. Yet only my post was thrown out to another thread as "off topic".
I thought that on XKCD we could have a fair and balanced discussion, but I guess I was mistaken. I'm outta here.
Other wildly offtopic posts since the modtext have simply been deleted. Would you have preferred that be my response to yours?
Re: Misunderstanding basic math concepts, help please?
Fluffy Intro
I really want to reply to so many of the points raised  lots of agreement, some disagreement, quite a bit of adding my own slant and a few misapprehensions about what I think (my fault for poorly expressing myself).
Unfortunately, a point by point addressing of the issues would make this post way too long. Plus I want to move things forward  while there are some interesting conversations to be had  they can easily get offtrack.
Sorry Demki  I really wasn't trying to ignore you, but you chimed in at a point where almost everyone was getting to the nitty gritty  Trying to keep a onetomany conversation going is challenging and I have to choose what I respond to  I get this choice wrong frequently  and sometimes I save up good arguments to be addressed later  I really do try to take note of everything people say  though I agree that this is not always obvious.
Overall I have most sympathy with the point of view expressed by Twistar  but as PsiCubed and Twistar noted, even where there is disagreement, there is a lot of common ground behind the scenes.
Regarding vague vs precise: There are many areas of mathematics (and, say, computers) where we can be incredibly precise in a reproducible manner. At the same time 'Cogito Ergo Sum' tells us we can never be certain about anything. Between these two points we have evidence of our own existence and the existence of structure that we can describe and reproduce in some detail.
While I agree that basic physics underlies emergent properties, it is also true that emergent properties are not very well handled by a purely reductionist and/or axiomatic approach to knowledge. (and just a note  mathematics doesn't have to be precise  statistics is a useful field...)
Boundaries, distinctions and observation
I put up a roadmap of where I think I might go. The first item on this roadmap was "Observation".
One of the things I have tried to do is apply rules (i.e. the Principle of Explosion) in ways that are strongly objected to. One of the principles of axiomatic mathematics is that each axiomatic system is independent of other axiomatic systems such that one inconsistent system does not infect other axiomatic systems with that inconsistency.
I strongly believe that it is not, in fact, possible to draw those distinctions. I believe that everything in this universe is fundamentally part of this universe and that there is no way to actually separate part of this universe, whether physically or logically.
I'm not sure how to make this argument convincing...
Putting aside 'many worlds interpretation', blackholes forming tunnels to elsewhere and whatever Dark Energy might be:
1. A universe is (by definition, I would argue) a self contained unit. (not unlike the desire for an individual axiomatic system to be a self contained unit).
While we can swap things around such that an electron spends some time looking like a pair of photons or vice versa; as far as we know, we can neither create nor destroy parts of our universe.
The Empty Set
In principle there might be many empty sets  but every empty set looks identical. The only method of distinguishing between sets is through their content. All sets that contain nothing have identical content. As such we have no means to tell one empty set apart from another empty set. As far as we are able to determine, each instance of an empty set is the same object as every other instance of the empty set.
Models, theories and descriptions outside our universe
Suppose we have some descriptions that describe things other than our universe (other universes, axiomatic systems outside our universe). If these models make no measurable difference to this universe then we have no way to distinguish a given model from all the other models that make no difference to this universe.
All models that are physically or logically distinct from this universe are equal and null with respect to this universe. (Asking what was 'before' or 'outside' this universe is generally regarded as a question without a meaningful answer).
Inconsistency
Within this universe it is impossible to do anything that is contrary to this universe. We cannot point to something within this universe and say "this universe does not permit that". (A Turing Machine can only emulate Turing Machines that it is capable of emulating  it cannot emulate Turing Machines that it cannot emulate).
Observation
In order for a mathematical model to be useful we need to observe that model (as well as describe it). When we consider Euclidean Geometry there is a physical process underlying that consideration. Whether photons carrying symbolic information to our eyes  or internal Neurons transmitting electrical signals  there is a physical connection between us, as observer, and the subject, no matter what the form of that subject.
This is Cogito Ergo Sum. We cannot interact with anything unless we are physically connected to the thing we are interacting with.
Whether expressed as a physical computer or an abstract/logical mathematical construction, we cannot do anything unless there is a physical connection between us and it.
If we have a well formed mathematical model  that model is of part of this universe. If it was not part of this universe it would tell us nothing that we can distinguish from all the other models that don't apply to this universe. If it is part of this universe then it is on exactly the same level as every other well formed model that is part of this universe.
Is Axiomatic Mathematics well formed given the above?
Specifically, exactly how is it possible to disconnect an axiomatic system from everything that allows us to describe, observe and manipulate that axiomatic system and still have that axiomatic system be relevant to all the things we just disconnected it from?
I really want to reply to so many of the points raised  lots of agreement, some disagreement, quite a bit of adding my own slant and a few misapprehensions about what I think (my fault for poorly expressing myself).
Unfortunately, a point by point addressing of the issues would make this post way too long. Plus I want to move things forward  while there are some interesting conversations to be had  they can easily get offtrack.
Sorry Demki  I really wasn't trying to ignore you, but you chimed in at a point where almost everyone was getting to the nitty gritty  Trying to keep a onetomany conversation going is challenging and I have to choose what I respond to  I get this choice wrong frequently  and sometimes I save up good arguments to be addressed later  I really do try to take note of everything people say  though I agree that this is not always obvious.
Overall I have most sympathy with the point of view expressed by Twistar  but as PsiCubed and Twistar noted, even where there is disagreement, there is a lot of common ground behind the scenes.
Regarding vague vs precise: There are many areas of mathematics (and, say, computers) where we can be incredibly precise in a reproducible manner. At the same time 'Cogito Ergo Sum' tells us we can never be certain about anything. Between these two points we have evidence of our own existence and the existence of structure that we can describe and reproduce in some detail.
While I agree that basic physics underlies emergent properties, it is also true that emergent properties are not very well handled by a purely reductionist and/or axiomatic approach to knowledge. (and just a note  mathematics doesn't have to be precise  statistics is a useful field...)
Boundaries, distinctions and observation
I put up a roadmap of where I think I might go. The first item on this roadmap was "Observation".
One of the things I have tried to do is apply rules (i.e. the Principle of Explosion) in ways that are strongly objected to. One of the principles of axiomatic mathematics is that each axiomatic system is independent of other axiomatic systems such that one inconsistent system does not infect other axiomatic systems with that inconsistency.
I strongly believe that it is not, in fact, possible to draw those distinctions. I believe that everything in this universe is fundamentally part of this universe and that there is no way to actually separate part of this universe, whether physically or logically.
I'm not sure how to make this argument convincing...
Putting aside 'many worlds interpretation', blackholes forming tunnels to elsewhere and whatever Dark Energy might be:
1. A universe is (by definition, I would argue) a self contained unit. (not unlike the desire for an individual axiomatic system to be a self contained unit).
While we can swap things around such that an electron spends some time looking like a pair of photons or vice versa; as far as we know, we can neither create nor destroy parts of our universe.
The Empty Set
In principle there might be many empty sets  but every empty set looks identical. The only method of distinguishing between sets is through their content. All sets that contain nothing have identical content. As such we have no means to tell one empty set apart from another empty set. As far as we are able to determine, each instance of an empty set is the same object as every other instance of the empty set.
Models, theories and descriptions outside our universe
Suppose we have some descriptions that describe things other than our universe (other universes, axiomatic systems outside our universe). If these models make no measurable difference to this universe then we have no way to distinguish a given model from all the other models that make no difference to this universe.
All models that are physically or logically distinct from this universe are equal and null with respect to this universe. (Asking what was 'before' or 'outside' this universe is generally regarded as a question without a meaningful answer).
Inconsistency
Within this universe it is impossible to do anything that is contrary to this universe. We cannot point to something within this universe and say "this universe does not permit that". (A Turing Machine can only emulate Turing Machines that it is capable of emulating  it cannot emulate Turing Machines that it cannot emulate).
Observation
In order for a mathematical model to be useful we need to observe that model (as well as describe it). When we consider Euclidean Geometry there is a physical process underlying that consideration. Whether photons carrying symbolic information to our eyes  or internal Neurons transmitting electrical signals  there is a physical connection between us, as observer, and the subject, no matter what the form of that subject.
This is Cogito Ergo Sum. We cannot interact with anything unless we are physically connected to the thing we are interacting with.
Whether expressed as a physical computer or an abstract/logical mathematical construction, we cannot do anything unless there is a physical connection between us and it.
If we have a well formed mathematical model  that model is of part of this universe. If it was not part of this universe it would tell us nothing that we can distinguish from all the other models that don't apply to this universe. If it is part of this universe then it is on exactly the same level as every other well formed model that is part of this universe.
Is Axiomatic Mathematics well formed given the above?
Specifically, exactly how is it possible to disconnect an axiomatic system from everything that allows us to describe, observe and manipulate that axiomatic system and still have that axiomatic system be relevant to all the things we just disconnected it from?

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Re: Misunderstanding basic math concepts, help please?
I can say only one thing: what you are describing here is not mathematics, it is some weird philosophy. If you are arguing using the physical universe, you aren't doing mathematics; mathematics is completely abstract.
One detail. You say: "In principle there might be many empty sets  but every empty set looks identical. The only method of distinguishing between sets is through their content. All sets that contain nothing have identical content. As such we have no means to tell one empty set apart from another empty set. As far as we are able to determine, each instance of an empty set is the same object as every other instance of the empty set." No, that's incorrect. Standard axiomatization of the set theory (ZFC) includes the axiom of extensionality, which says that a set is fully defined by its elements. If sets A and B have exactly the same elements, they aren't kinda sorta equal, they aren't just indistinguishable; they are the same set. Period.
One detail. You say: "In principle there might be many empty sets  but every empty set looks identical. The only method of distinguishing between sets is through their content. All sets that contain nothing have identical content. As such we have no means to tell one empty set apart from another empty set. As far as we are able to determine, each instance of an empty set is the same object as every other instance of the empty set." No, that's incorrect. Standard axiomatization of the set theory (ZFC) includes the axiom of extensionality, which says that a set is fully defined by its elements. If sets A and B have exactly the same elements, they aren't kinda sorta equal, they aren't just indistinguishable; they are the same set. Period.
Re: Misunderstanding basic math concepts, help please?
I'm out. Treatid, take an actual math class. I suggest "real analysis".
Tons of online courses in math, science and other fields
Tons of online courses in math, science and other fields
Re: Misunderstanding basic math concepts, help please?
Oh dear  it seems I have miscommunicated again.
I was not trying to rewrite set theory or say anything new about The Empty Set.
I was trying to point to an existing precedent that says that things that are indistinguishable are one thing.
...
Referencing this universe is no different to referencing any other mathematical system. I was under the impression that most people here think that we can use mathematics to describe this universe. As such, we can think of this universe in a similar way to any other mathematical system  perhaps an axiomatic system or a Turing Machine...
The only difference is that I have made the observer an explicit part of the system being observed.
...
I am well aware that mathematics tries to draw a line between different axiomatic systems. I have been told that this is the case many times.
What I haven't seen is a clear description of how that line is drawn.
Exactly how do axioms defined in a natural language suddenly become not defined in a natural language but still retain their meaning?
Where exactly is the sharp transition from natural language to formal language when the latter depends on the former in order to be bootstrapped into existence?
I can easily understand you being tired of me. I can easily understand miscommunication from me to you.
But I get the sense that you have (again) decided that I don't understand logic because... what?
I was not trying to rewrite set theory or say anything new about The Empty Set.
I was trying to point to an existing precedent that says that things that are indistinguishable are one thing.
...
Referencing this universe is no different to referencing any other mathematical system. I was under the impression that most people here think that we can use mathematics to describe this universe. As such, we can think of this universe in a similar way to any other mathematical system  perhaps an axiomatic system or a Turing Machine...
The only difference is that I have made the observer an explicit part of the system being observed.
...
I am well aware that mathematics tries to draw a line between different axiomatic systems. I have been told that this is the case many times.
What I haven't seen is a clear description of how that line is drawn.
Exactly how do axioms defined in a natural language suddenly become not defined in a natural language but still retain their meaning?
Where exactly is the sharp transition from natural language to formal language when the latter depends on the former in order to be bootstrapped into existence?
I can easily understand you being tired of me. I can easily understand miscommunication from me to you.
But I get the sense that you have (again) decided that I don't understand logic because... what?
 gmalivuk
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Re: Misunderstanding basic math concepts, help please?
Because you've been making the same mistakes for years and people have been trying to explain things to you for that same amount of time.Treatid wrote:But I get the sense that you have (again) decided that I don't understand logic because... what?
Re: Misunderstanding basic math concepts, help please?
I'm really not trying to be obtuse here.
That one axiomatic system is distinct from another axiomatic system seems to me to be a really important part of axiomatic mathematics. I can see that the distinction is essential. I understand that the distinction is an assumed part of what axiomatic mathematics is.
And I agree that I have been hammering on this point for a long time.
But for all the time that people have been explaining the importance of this distinction  I don't feel they have been showing me that the distinction can actually exist. (it needs to exist  but I haven't been shown the specific mechanism that allows it to exist).
In practice, in this universe, we cannot completely separate one part of this universe from another part of this universe.
You (mathematicians) tell me that axiomatic mathematics can split off pieces of logic from other pieces of logic. And I understand the desire for isolating parts of a problem so they are tractable  solving everything in one go is difficult  so split up the problem into isolated parts.
...
Starting with no language, no universe... starting from nothing whatsoever  axiomatic mathematics cannot pull itself up by its boot straps. Axiomatic mathematics relies on something else (in this case, natural language) to get started.
If you succeed in actually showing an absolute distinction between natural language and axiomatic mathematics  then surely you have lost that essential bootstrap?
So which is it? A) Axiomatic mathematics relies on natural language to get started OR B) axiomatic systems are entirely distinct from natural language?
Or perhaps it is C) There is a degree of separation between natural language and individual axiomatic systems  but not absolute separation?
...
I can create a Virtual Machine on my computer. That Virtual Machine is largely distinct from my computer. I can create a Virtual Machine within that Virtual Machine. That new Virtual Machine is also largely distinct from both my computer and the first Virtual Machine. I can potentially keep nesting Virtual Machines.
So... What happens to a Virtual Machine 20 layers of separation down when I turn off my computer? (Answer: no matter how deeply nested  no matter how virtual  that last Virtual Machine is inextricably reliant on the existence of my computer. My computer turns off and the Virtual Machine also switches off).
...
So  what, exactly, is the difference between natural language booting up axiomatic mathematics and my computer booting up a Virtual Machine?
I have no doubt that you feel this exact question has been answered multiple times.
From my perspective  I've been told that there is a distinction multiple times. But I haven't yet been shown that such a distinction exists or can exist as an absolute.
(Note that I'm not saying there is no difference  I do agree that there is a difference between formal and informal languages. I'm saying that the difference is not absolute. It is merely a degree of difference, as opposed to an absolute distinction. Perhaps we are talking past one another  perhaps you think that a degree of difference is sufficient? Perhaps you think insisting on an absolute difference is unreasonable?  just as setting the bar for absolute knowledge as... absolute... has been argued as being unreasonable?)
That one axiomatic system is distinct from another axiomatic system seems to me to be a really important part of axiomatic mathematics. I can see that the distinction is essential. I understand that the distinction is an assumed part of what axiomatic mathematics is.
And I agree that I have been hammering on this point for a long time.
But for all the time that people have been explaining the importance of this distinction  I don't feel they have been showing me that the distinction can actually exist. (it needs to exist  but I haven't been shown the specific mechanism that allows it to exist).
In practice, in this universe, we cannot completely separate one part of this universe from another part of this universe.
You (mathematicians) tell me that axiomatic mathematics can split off pieces of logic from other pieces of logic. And I understand the desire for isolating parts of a problem so they are tractable  solving everything in one go is difficult  so split up the problem into isolated parts.
...
Starting with no language, no universe... starting from nothing whatsoever  axiomatic mathematics cannot pull itself up by its boot straps. Axiomatic mathematics relies on something else (in this case, natural language) to get started.
If you succeed in actually showing an absolute distinction between natural language and axiomatic mathematics  then surely you have lost that essential bootstrap?
So which is it? A) Axiomatic mathematics relies on natural language to get started OR B) axiomatic systems are entirely distinct from natural language?
Or perhaps it is C) There is a degree of separation between natural language and individual axiomatic systems  but not absolute separation?
...
I can create a Virtual Machine on my computer. That Virtual Machine is largely distinct from my computer. I can create a Virtual Machine within that Virtual Machine. That new Virtual Machine is also largely distinct from both my computer and the first Virtual Machine. I can potentially keep nesting Virtual Machines.
So... What happens to a Virtual Machine 20 layers of separation down when I turn off my computer? (Answer: no matter how deeply nested  no matter how virtual  that last Virtual Machine is inextricably reliant on the existence of my computer. My computer turns off and the Virtual Machine also switches off).
...
So  what, exactly, is the difference between natural language booting up axiomatic mathematics and my computer booting up a Virtual Machine?
I have no doubt that you feel this exact question has been answered multiple times.
From my perspective  I've been told that there is a distinction multiple times. But I haven't yet been shown that such a distinction exists or can exist as an absolute.
(Note that I'm not saying there is no difference  I do agree that there is a difference between formal and informal languages. I'm saying that the difference is not absolute. It is merely a degree of difference, as opposed to an absolute distinction. Perhaps we are talking past one another  perhaps you think that a degree of difference is sufficient? Perhaps you think insisting on an absolute difference is unreasonable?  just as setting the bar for absolute knowledge as... absolute... has been argued as being unreasonable?)

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Re: Misunderstanding basic math concepts, help please?
Treatid wrote:So which is it? A) Axiomatic mathematics relies on natural language to get started OR B) axiomatic systems are entirely distinct from natural language?
A. No question. The answer is A. We've said this many, many times.
Re: Misunderstanding basic math concepts, help please?
Treatid wrote:...
I can create a Virtual Machine on my computer. That Virtual Machine is largely distinct from my computer. I can create a Virtual Machine within that Virtual Machine. That new Virtual Machine is also largely distinct from both my computer and the first Virtual Machine. I can potentially keep nesting Virtual Machines.
So... What happens to a Virtual Machine 20 layers of separation down when I turn off my computer? (Answer: no matter how deeply nested  no matter how virtual  that last Virtual Machine is inextricably reliant on the existence of my computer. My computer turns off and the Virtual Machine also switches off).
...
So  what, exactly, is the difference between natural language booting up axiomatic mathematics and my computer booting up a Virtual Machine?
Not a lot.
In fact, it's a pretty good analogy.
Suppose your machine has a bug in the OS that means that certain calculations give wrong answers. By the time you've made it down 20 layers, will that VM also give the wrong answers to those calculations? Only if they get passed through the entire chain of reinterpretation to be the same calculations by the time they reach the original OS. All it takes is for one of the VMs in the stack to convert the calculation into something else, or to have a deliberate patch for the error, and the bug ceases to be a problem.
Particularly if the VMs have a wide range of architectures, you can reach a point where it makes very little difference (except in terms of speed of operation) what the actual physical machine or the root OS is  and there are hardly any bugs that can actually affect the final behaviour of the deepest VM.
Mathematics can run just fine on pretty much any natural language because the few features it absolutely relies on are (probably) universal ones  and the choice of natural language has minimal effect on the mathematics that can be done on top of it.
 Soupspoon
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Re: Misunderstanding basic math concepts, help please?
The best Real World version of that analogy might be the old original Pentium floating point bug^{1}. It would show up (if you knew what you were looking for) in a simple Excel formula, but if you implemented a 'manual' division in a 'metacalculation' of some arbitrary but integerbased calculation then you could avoid it. (At the expense of the inefficiency of layering.)
Not directly relevent, perhaps, but came to mind anyway...
^{1} Triggered in a certain circumstance (as trivial as a 5_{hex} digit in a particular position of value in a particular type of division sent to the FPU, IIRC), it could be encountered in many trivial floatingpoint circumstances, the saving grace (probably) is that it is just a very small subset of potential operations. It was probably invoked far more by deliberate testing/discovery than ever would happened if it was just quietly fixed in the next iteration of processor die like Intel probably wanted.
Not directly relevent, perhaps, but came to mind anyway...
^{1} Triggered in a certain circumstance (as trivial as a 5_{hex} digit in a particular position of value in a particular type of division sent to the FPU, IIRC), it could be encountered in many trivial floatingpoint circumstances, the saving grace (probably) is that it is just a very small subset of potential operations. It was probably invoked far more by deliberate testing/discovery than ever would happened if it was just quietly fixed in the next iteration of processor die like Intel probably wanted.
Re: Misunderstanding basic math concepts, help please?
Furiously Agree with each other
It looks like it was a mistake to try and move things forward without explicitly acknowledging the points that I agree or disagree with:
We've already established that we cannot specify axioms with regard to 1) in such a way that proofs depending from those axioms are "absolute epistemic certainty". We can, at least locally, work out whether a given number is prime and be confident that other mathematicians will recognise what we mean by 'number', 'prime' and agree on which numbers are prime and which not prime.
On the other hand, we know that there are many legitimate definitions of a Number Line  some Number Lines are open, Some closed (meet at +/ infinity). A Number Line on a sphere (embedded in Euclidean space) will read differently if we read around the sphere vs a (Euclidean) straight line between points.
2+2=4 is not remotely universal. It just so happens that a flat Euclidean space is a good approximation for many of the practical things we find useful. (And there is a pretty good chance that other beings that have evolved in a similar way on a similar planet are also biased in favour of a similar set of assumptions). So  widespread  and so common to people that it is easy to mistake for universal among Earthlings  but not universal in some absolute sense such that any being would automatically understand the symbols and have made identical associations.
Cleverbeans provided Gödel's incompleteness theorems link pertaining to 2). It is worth noting that the link references (among other important limitations) Tarski's undefinability theorem that (loosely) says we cannot define 'true' within a system (the proof is specific to arithmetic  but can be generalised, with care).
Which brings us to 3)  Metamathematics.
This is the point I was trying to address in my most recent posts. The link to Gödel's incompleteness theorems also references various other fundamental limitations of mathematics. It is my perception that mathematicians go to some effort to find loopholes around these limitations. One group of these loopholes involve some form of metamathematics in which one system references another system thereby overcoming limitations of selfreference within a system.
As I hope is clear, I think it is impossible to start with one system and suddenly arrive at some other system that is actually distinct and separate from the preceding system. I think there must always be a fundamental connection between a describing system and a described system (or an observing system and an observed system).
Even if we take a typical computer as a Universal Turing Machine capable of emulating any other Turing Machine, those subsequent emulated machines (Virtual Machines) are still fundamentally a part of the root computer  even when the emulation is of an apparently different instruction set, switching off the root computer will switch off all the emulated computers. (I would also argue that designing of those emulation layers cannot be done in isolation  but for the moment I'm only arguing that there is a direct connection between the layers  not the degree of that connection).
...
Arbiteroftruth: yes (I agree with you)  it is absolutely clear that axiomatic mathematics needs natural language to get started. It has also been made absolutely clear that the rules for natural languages are utterly distinct from the rules for formal systems such that the Principle of Explosion absolutely definitely applies to one and absolutely definitely does not apply to the other.
Similarly, Each and every axiomatic system descends from natural language at some degree of remove  but each and every axiomatic system is definitely distinct from every other axiomatic system such that an inconsistent axiomatic system has no relationship and thus no impact on any other (hopefully consistent) axiomatic system.
It isn't that people aren't clear  it is that these various things that people are utterly clear about appear (to me) to contradict each other.
rmsgrey recognises that there is a chain of connection such that some things follow through from beginning to end (e.g. the power supply to the physical computer). The point regarding emulation of distinct instruction sets is also important  but one I'd like to come back to later.
It looks like it was a mistake to try and move things forward without explicitly acknowledging the points that I agree or disagree with:
Twistar & gmaliveuk wrote:edit: Oh and another important point. We identified four types of mathematical questions:
1) Questions about the truthhood of certain mathematical sentences which can be proven from the axioms of mathematics. Questions like "is 85 a prime number?"
2) Questions about the truthhood of certain mathematical sentences which CANNOT be proven from the axioms of mathematics and which are independent of those axioms. This means that that mathematical sentence could be taken to be either true or false and neither would lead to a contradiction. The vocab word here is "independent". An example is the continuum hypothesis.
3) Questions about the truthhood of certain mathematical sentences which CANNOT be proven from the axioms of mathematics but which CAN be proven using metamathematics. The Godel sentence is the obvious example here but maybe Gmalivuk's questions about whether certain real numbers are normal also falls in this category.
3a) I suspect some questions about uncomputable numbers can't be proven metamathematically, either, as they would imply solutions to the halting problem. Perhaps call those questions type 3a and leave undetermined whether 3a=3
4) Questions which are not mathematical questions at all. Things like "is Euler's formula beautiful." And don't give me some bullshit about mathematical definitions of beauty because a) you're just being obnoxious and you know it and b) I'll just come up with a more unrelated example and you know that too.
We've already established that we cannot specify axioms with regard to 1) in such a way that proofs depending from those axioms are "absolute epistemic certainty". We can, at least locally, work out whether a given number is prime and be confident that other mathematicians will recognise what we mean by 'number', 'prime' and agree on which numbers are prime and which not prime.
On the other hand, we know that there are many legitimate definitions of a Number Line  some Number Lines are open, Some closed (meet at +/ infinity). A Number Line on a sphere (embedded in Euclidean space) will read differently if we read around the sphere vs a (Euclidean) straight line between points.
2+2=4 is not remotely universal. It just so happens that a flat Euclidean space is a good approximation for many of the practical things we find useful. (And there is a pretty good chance that other beings that have evolved in a similar way on a similar planet are also biased in favour of a similar set of assumptions). So  widespread  and so common to people that it is easy to mistake for universal among Earthlings  but not universal in some absolute sense such that any being would automatically understand the symbols and have made identical associations.
Cleverbeans provided Gödel's incompleteness theorems link pertaining to 2). It is worth noting that the link references (among other important limitations) Tarski's undefinability theorem that (loosely) says we cannot define 'true' within a system (the proof is specific to arithmetic  but can be generalised, with care).
Which brings us to 3)  Metamathematics.
This is the point I was trying to address in my most recent posts. The link to Gödel's incompleteness theorems also references various other fundamental limitations of mathematics. It is my perception that mathematicians go to some effort to find loopholes around these limitations. One group of these loopholes involve some form of metamathematics in which one system references another system thereby overcoming limitations of selfreference within a system.
As I hope is clear, I think it is impossible to start with one system and suddenly arrive at some other system that is actually distinct and separate from the preceding system. I think there must always be a fundamental connection between a describing system and a described system (or an observing system and an observed system).
Even if we take a typical computer as a Universal Turing Machine capable of emulating any other Turing Machine, those subsequent emulated machines (Virtual Machines) are still fundamentally a part of the root computer  even when the emulation is of an apparently different instruction set, switching off the root computer will switch off all the emulated computers. (I would also argue that designing of those emulation layers cannot be done in isolation  but for the moment I'm only arguing that there is a direct connection between the layers  not the degree of that connection).
...
Arbiteroftruth: yes (I agree with you)  it is absolutely clear that axiomatic mathematics needs natural language to get started. It has also been made absolutely clear that the rules for natural languages are utterly distinct from the rules for formal systems such that the Principle of Explosion absolutely definitely applies to one and absolutely definitely does not apply to the other.
Similarly, Each and every axiomatic system descends from natural language at some degree of remove  but each and every axiomatic system is definitely distinct from every other axiomatic system such that an inconsistent axiomatic system has no relationship and thus no impact on any other (hopefully consistent) axiomatic system.
It isn't that people aren't clear  it is that these various things that people are utterly clear about appear (to me) to contradict each other.
rmsgrey recognises that there is a chain of connection such that some things follow through from beginning to end (e.g. the power supply to the physical computer). The point regarding emulation of distinct instruction sets is also important  but one I'd like to come back to later.
Re: Misunderstanding basic math concepts, help please?
If I'm running two different VMs on my computer at the same time, and one crashes, it doesn't mean that the other has to go down with it. In fact, the VMs probably don't interact with each other at all. This is the way that different axiomatic systems are separate.
If two different computers run the same VM, then ideally the two instances of the VM run exactly the same. So in a big way, the VM is independent of the machine that runs it (even if each instance of it is tied to a computer). This is how an axiomatic system is separate from a system it describes and reasons about.
I hope the analogies are clear.
If two different computers run the same VM, then ideally the two instances of the VM run exactly the same. So in a big way, the VM is independent of the machine that runs it (even if each instance of it is tied to a computer). This is how an axiomatic system is separate from a system it describes and reasons about.
I hope the analogies are clear.
(∫p^{2})(∫q^{2}) ≥ (∫pq)^{2}
Thanks, skeptical scientist, for knowing symbols and giving them to me.
Thanks, skeptical scientist, for knowing symbols and giving them to me.
Re: Misunderstanding basic math concepts, help please?
If you completely lose the ability to use natural language (the closest analogue I can think of to turning off the root computer) then, sure, all of mathematics goes down with it. but that doesn't mean that every flaw of natural language is inherited by mathematics any more than the fact pulling the plug axes VMs means running the VMs on a first generation pentium means the VMs will exhibit the floating point bug.
Re: Misunderstanding basic math concepts, help please?
I want to take my time about responding to Cauchy and rmsgrey.
What they are saying is accurate  but is missing (very reasonably) the point I'm (trying to) make. I think that they, and axiomatic mathematics, are missing how important the difference between "absolutely distinct in every conceivable way" and "mostly separate, but not quite entirely so" is.
At the moment, my perception is that axiomatic mathematics is having its cake and eating it by pretending to have absolute distinctions but then not behaving in accordance with that pretence.
A bit of Housekeeping
In my previous post I conflated Twistar's 2) "statements independent of the axioms" with "true with respect to the axioms but not descending directly from the axioms". Not immediately critical  but should be an independent entry 1b) within the list of types of mathematical questions/answers.
The main event  Degree of difference
Way back when, we had the classic laws of thought, and, in particular, the law of the excluded middle. Most people objected to me introducing this as a foundational principle of mathematics. They further objected to me trying to use that principle to show that mathematics reasoning that relied on an included middle was trying to have your cake and eat it  you either are excluding the middle, or you are not.
As such, when we say that one Virtual Machine (Axiomatic System) is distinct from another Virtual Machine in the context of mathematics I take that to mean "distinct in an absolute sense." "No connection whatsoever." "Utterly unconnected in every conceivable way."
It is, to me, the difference between saying "x and !x are entirely separate" and saying "x and !x are different but have some overlap with each other".
If you permit that there is even the slightest, most indirect connection between one axiomatic system and another axiomatic system, then, it appears to me, you are also admitting that a venn diagram of x and !x has an overlapping section where something is both x and !x to some degree.
No absolutes
To the degree that it has been conceded that there does not exist an "absolute epistemic certainty", it is also conceded that we cannot know that the venn diagram of x and !x has no overlap.
A closed system
I linked Gödel's incompleteness theorems and the related link Tarski's undefinability theorem last post.
Those pages describe and link to a number of fundamental limitations that apply to closed systems. Mathematics has already been down this road. There are already recognised proofs that a single closed system cannot do various things that mathematicians wave away with metamathematics and 'distinct systems' which despite being distinct are somehow able to make meaningful statements about systems that they are supposedly unconnected to.
Partial Descriptions
We also spent some time considering partial descriptions wherein even where we are certain about the bits we have described  the unknown bits, the undescribed bits could be anything. The unknown, by virtue of being unknown, is unconstrained in the ways that it can impact on the known. Unless we find a different approach to knowledge, any degree of doubt or uncertainty renders everything we think we know entirely contingent on that doubt and uncertainty.
(Which, of course, is a large part of why each of the 3 classic laws of thought were established. This isn't new reasoning. You either hold entirely and absolutely to the laws of thought  or you (apparently) give up all possibility of saying anything meaningful).
(Obviously we can say things that are meaningful to us  so obviously something else is at play  but if mathematics cannot show that x and !x have no overlap, then huge chunks of mathematics lose their power to describe).
Barely any difference
Two different Virtual Machines running different instruction sets, operating systems and software on the same computer are very different from one another. The similarity between them is so tiny as to appear pretty much insignificant.
The trouble is that the difference between 'x and !x' being absolute versus partial is critical to huge amounts of mathematics. If there is even the tiniest amount that x = !x then mathematical logic collapses (I know 'partially equals' is a misuse of '='  but that distinction is the subject, so...). The excluded middle (along with the other laws of thought which each have the same goal  to specify absolutes, whether of identity or distinction) are essential to the functioning of mathematics. You cannot throw away the absolute distinction between x and !x and carry on as if it doesn't matter (well  "shoudln't be able to"...).
Virtual Machines
Cauchy and rmsgrey provide a counter argument to all of the above. "The separation between Virtual Machines may not be absolute in some really pedantic sense  but it is good enough that we can treat the Virtual Machines as if they were actually distinct in an absolute sense."
"Sure  there are some edge cases where if the power goes out in the neighbourhood, or a cosmic ray causes a physical processor to glitch  but those are unimportant edge cases."
Edge cases
What is it you think I've been arguing all this time? Surely by now you realise that I'm a pedantic son of a bitch (sorry mum).
Granted  axiomatic mathematics is a good rule of thumb that is practically useful for building bridges that don't fall down.
But when we get really perversely pedantic about it... there is no "absolute epistemic certainty" that x is utterly distinct from !x for any x (let alone for all x).
So long as we apply (axiomatic) mathematics to real world examples where the difference between x and !x is significantly more important than the similarity then mathematics works.
And where the distinction between x and !x is less clear cut, then (axiomatic) mathematics fails. In this respect, CleverBeans' examples are absolutely pertinent and on point. Subjective qualia are one of the areas where we cannot draw a hard distinction  because there is no hard distinction.
x and !x
Right now, noone can prove from first principles, beyond any doubt, to the standard of "absolute epistemic knowledge" that there exists anything, anywhere that is definitely, totally its own thing and definitely and totally not part of something else.
Indeed, the way we define the meaning of words  by the use of other words  strongly suggests that the meaning for all words (at least) are inextricably linked to, and part of, the meaning of other words. It isn't possible to take a word (set of symbols) completely out of context and still be certain what that word means.
This is the direction of my overall argument: meaning can only derive from the full context. Nothing has any meaning by itself (see the argument above regarding partial description). Even the teeny, tiny, weeniest of gaps in the full context leaves us with an unknown element capable of rendering everything we think we have described irrelevant.
What they are saying is accurate  but is missing (very reasonably) the point I'm (trying to) make. I think that they, and axiomatic mathematics, are missing how important the difference between "absolutely distinct in every conceivable way" and "mostly separate, but not quite entirely so" is.
At the moment, my perception is that axiomatic mathematics is having its cake and eating it by pretending to have absolute distinctions but then not behaving in accordance with that pretence.
A bit of Housekeeping
In my previous post I conflated Twistar's 2) "statements independent of the axioms" with "true with respect to the axioms but not descending directly from the axioms". Not immediately critical  but should be an independent entry 1b) within the list of types of mathematical questions/answers.
The main event  Degree of difference
Way back when, we had the classic laws of thought, and, in particular, the law of the excluded middle. Most people objected to me introducing this as a foundational principle of mathematics. They further objected to me trying to use that principle to show that mathematics reasoning that relied on an included middle was trying to have your cake and eat it  you either are excluding the middle, or you are not.
As such, when we say that one Virtual Machine (Axiomatic System) is distinct from another Virtual Machine in the context of mathematics I take that to mean "distinct in an absolute sense." "No connection whatsoever." "Utterly unconnected in every conceivable way."
It is, to me, the difference between saying "x and !x are entirely separate" and saying "x and !x are different but have some overlap with each other".
If you permit that there is even the slightest, most indirect connection between one axiomatic system and another axiomatic system, then, it appears to me, you are also admitting that a venn diagram of x and !x has an overlapping section where something is both x and !x to some degree.
No absolutes
To the degree that it has been conceded that there does not exist an "absolute epistemic certainty", it is also conceded that we cannot know that the venn diagram of x and !x has no overlap.
A closed system
I linked Gödel's incompleteness theorems and the related link Tarski's undefinability theorem last post.
Those pages describe and link to a number of fundamental limitations that apply to closed systems. Mathematics has already been down this road. There are already recognised proofs that a single closed system cannot do various things that mathematicians wave away with metamathematics and 'distinct systems' which despite being distinct are somehow able to make meaningful statements about systems that they are supposedly unconnected to.
Partial Descriptions
We also spent some time considering partial descriptions wherein even where we are certain about the bits we have described  the unknown bits, the undescribed bits could be anything. The unknown, by virtue of being unknown, is unconstrained in the ways that it can impact on the known. Unless we find a different approach to knowledge, any degree of doubt or uncertainty renders everything we think we know entirely contingent on that doubt and uncertainty.
(Which, of course, is a large part of why each of the 3 classic laws of thought were established. This isn't new reasoning. You either hold entirely and absolutely to the laws of thought  or you (apparently) give up all possibility of saying anything meaningful).
(Obviously we can say things that are meaningful to us  so obviously something else is at play  but if mathematics cannot show that x and !x have no overlap, then huge chunks of mathematics lose their power to describe).
Barely any difference
Two different Virtual Machines running different instruction sets, operating systems and software on the same computer are very different from one another. The similarity between them is so tiny as to appear pretty much insignificant.
The trouble is that the difference between 'x and !x' being absolute versus partial is critical to huge amounts of mathematics. If there is even the tiniest amount that x = !x then mathematical logic collapses (I know 'partially equals' is a misuse of '='  but that distinction is the subject, so...). The excluded middle (along with the other laws of thought which each have the same goal  to specify absolutes, whether of identity or distinction) are essential to the functioning of mathematics. You cannot throw away the absolute distinction between x and !x and carry on as if it doesn't matter (well  "shoudln't be able to"...).
Virtual Machines
Cauchy and rmsgrey provide a counter argument to all of the above. "The separation between Virtual Machines may not be absolute in some really pedantic sense  but it is good enough that we can treat the Virtual Machines as if they were actually distinct in an absolute sense."
"Sure  there are some edge cases where if the power goes out in the neighbourhood, or a cosmic ray causes a physical processor to glitch  but those are unimportant edge cases."
Edge cases
What is it you think I've been arguing all this time? Surely by now you realise that I'm a pedantic son of a bitch (sorry mum).
Granted  axiomatic mathematics is a good rule of thumb that is practically useful for building bridges that don't fall down.
But when we get really perversely pedantic about it... there is no "absolute epistemic certainty" that x is utterly distinct from !x for any x (let alone for all x).
So long as we apply (axiomatic) mathematics to real world examples where the difference between x and !x is significantly more important than the similarity then mathematics works.
And where the distinction between x and !x is less clear cut, then (axiomatic) mathematics fails. In this respect, CleverBeans' examples are absolutely pertinent and on point. Subjective qualia are one of the areas where we cannot draw a hard distinction  because there is no hard distinction.
x and !x
Right now, noone can prove from first principles, beyond any doubt, to the standard of "absolute epistemic knowledge" that there exists anything, anywhere that is definitely, totally its own thing and definitely and totally not part of something else.
Indeed, the way we define the meaning of words  by the use of other words  strongly suggests that the meaning for all words (at least) are inextricably linked to, and part of, the meaning of other words. It isn't possible to take a word (set of symbols) completely out of context and still be certain what that word means.
This is the direction of my overall argument: meaning can only derive from the full context. Nothing has any meaning by itself (see the argument above regarding partial description). Even the teeny, tiny, weeniest of gaps in the full context leaves us with an unknown element capable of rendering everything we think we have described irrelevant.

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Re: Misunderstanding basic math concepts, help please?
Yes. And that's why mathematicians don't really bother with this "meaning" thing. Actually, that's not really true: they try, but it can get subjective and they often feel misunderstood. Like everyone else who attempts to talk about anything "in full context".Treatid wrote:This is the direction of my overall argument: meaning can only derive from the full context. Nothing has any meaning by itself (see the argument above regarding partial description).
Axiomatic mathematics isn't an attempt to provide a "full context". It's only a tool that proved to be very useful because:
1) Unlike proofs that aren't formal, people generally agree on whether or not a formal proof is "correct" (in the fuzzy natural language sense). Formal proofs are "objective" (in the fuzzy natural language sense) because people are good (in the sense that they generally agree and expect anyone who disagrees to be incompetent or mistaken) at checking whether or not symbolic manipulations fit the one the axioms "allow" (axioms are stated in fuzzy natural language, but we act like this part of fuzzy natural language is sufficiently free of trouble).
2) People generally feel that, when talking about numbers and stuff, nothing is lost when translating from the subjective "full context" universe of their minds to symbolic manipulations. When talking about, say, modular arithmetic, people generally feel that they can say everything they want to say using the standard formalism after getting some familiarity with the axioms. This is surprising. It seems to me that for some topics in higher mathematics, people can get more uncomfortable with some axioms, but in those cases they tend to be very mindful of which axioms are being used, and exploring the effects of starting from different axioms is part of the point.
Again, this is not a metaphysical justification for axiomatic mathematics. It's just an explanation of why it is used. Some people like to argue that mathematics is "true" in some metaphysical sense. Of course, since no one can prove anything to the standard of "absolute epistemic knowledge", we don't know. But we don't really care that much because the utility we perceive in using an axiomatic formalism is a very empirical thing, it doesn't really come from some epistemic ego trip.
Re: Misunderstanding basic math concepts, help please?
Treatid wrote:I think that they, and axiomatic mathematics, are missing how important the difference between "absolutely distinct in every conceivable way" and "mostly separate, but not quite entirely so" is.
What does it mean for two systems to be "mostly separate, but not quite entirely so"?
As such, when we say that one Virtual Machine (Axiomatic System) is distinct from another Virtual Machine in the context of mathematics I take that to mean "distinct in an absolute sense." "No connection whatsoever." "Utterly unconnected in every conceivable way."
Do you know what distinct means? It means "different" i.e. "not the same".
It is, to me, the difference between saying "x and !x are entirely separate" and saying "x and !x are different but have some overlap with each other".
What kind of things are x and !x such that they can have "some overlap"? Because sentences can't overlap, that doesn't even make sense.
If you permit that there is even the slightest, most indirect connection between one axiomatic system and another axiomatic system, then, it appears to me, you are also admitting that a venn diagram of x and !x has an overlapping section where something is both x and !x to some degree.
So x is a system? What is the negation of a system? I feel like we talked about this earlier, and you kind of dodged the question. Remember, the Principle of Explosion is a thing that happens inside a given system. If x and !x are statements in a system, why do other systems matter? If they're anything else, why does the Principle of Explosion apply to them?
To the degree that it has been conceded that there does not exist an "absolute epistemic certainty", it is also conceded that we cannot know that the venn diagram of x and !x has no overlap.
What is in the Venn diagram of x? Do things like the collection of symbols go in the Venn diagram? The deductive rules?
Your entire argument seems to hinge on this "partial overlap" between these nebulous things x and !x. Until you clear things up (and maybe after then, too) I don't think your argument holds water.
(∫p^{2})(∫q^{2}) ≥ (∫pq)^{2}
Thanks, skeptical scientist, for knowing symbols and giving them to me.
Thanks, skeptical scientist, for knowing symbols and giving them to me.
Re: Misunderstanding basic math concepts, help please?
Treadid: I think I can summarize your entire problem with mathematics as the following:
 Maths is communicated between human beings
 Communication between human beings is fuzzy with the potential for error
 Therefore maths is fuzzy with the potential for error
But, so what? Noone ever claimed that mathematical proofs as carried out by human beings are always free of error. Indeed, errors creep in all the time and it can take months for novel and complex proofs to be peer reviewed.
But the beauty of maths is that it's like a box of lego that has relatively few relatively simple kinds of blocks. Even though you might built a model as tall as a skyscraper, it's relatively easy for large numbers of people to verify that only permitted blocks have been used and they have been slotted together in only the permitted fashion. And while a subtle error might be overlooked for a long time, the best thing about maths is that once someone spots it, it's quite easy for others to confirm it.
Is mass delusion possible? Is there a glaring fundamental flaw in maths that, just by pure chance, noone has ever spotted? Sure, you can't rule it out absolutely. But, to the extent that we can rely on anything in this universe, we can rely upon mathematics.
 Maths is communicated between human beings
 Communication between human beings is fuzzy with the potential for error
 Therefore maths is fuzzy with the potential for error
But, so what? Noone ever claimed that mathematical proofs as carried out by human beings are always free of error. Indeed, errors creep in all the time and it can take months for novel and complex proofs to be peer reviewed.
But the beauty of maths is that it's like a box of lego that has relatively few relatively simple kinds of blocks. Even though you might built a model as tall as a skyscraper, it's relatively easy for large numbers of people to verify that only permitted blocks have been used and they have been slotted together in only the permitted fashion. And while a subtle error might be overlooked for a long time, the best thing about maths is that once someone spots it, it's quite easy for others to confirm it.
Is mass delusion possible? Is there a glaring fundamental flaw in maths that, just by pure chance, noone has ever spotted? Sure, you can't rule it out absolutely. But, to the extent that we can rely on anything in this universe, we can rely upon mathematics.
Re: Misunderstanding basic math concepts, help please?
What does !x mean in the notation being used by Treatid? "Notx" could mean "the direct opposite of x", "the complement of x", "some arbitrary thing that isn't identically equal to x" or a couple of other variations.
Re: Misunderstanding basic math concepts, help please?
Cauchy: I hope you are also a teacher outside these forums. Even when you think people are wrong  you don't tell them they are wrong, you give them a carefully measured piece of rope exquisitely judged to let them hang themselves (realise where they have made a mistake). Your consistent constructive involvement has my immense respect.
(not criticising anyone else  just that Cauchy is really, really good  not just with me  he seems to have a real talent for cutting through confusion in all the threads I've seen him in).
Moiraemachy & elasto: Yes, I strongly agree with the majority of what you say. And if that was the end of the story I would shutup and accept axiomatic mathematics, warts, limitations and even contradictions, as the best we have.
However, I think there is an alternative that avoids some of the problems of axiomatic mathematics.
I'd like to describe that alternative.
The trouble is that I think there is a similar thing to Euclidean Geometry before it was realised that the fifth postulate was distinct from the previous four. Or how General Relativity is sufficiently different from Newtonian Mechanics that there isn't a direct transform between the two.
As such, when I try to describe the alternative  it looks wrong from the perspective of axiomatic thought. I'm not attacking axiomatic mathematics because I have taken a dislike to it  I'm trying to undermine some of the assumptions behind axiomatic mathematics because they appear to be getting in the way of seeing what an alternative to axiomatic mathematics might be.
For example, axiomatic mathematics tries to avoid the inherent fuzziness and uncertainty in human communication. But that fuzziness and uncertainty is a fundamental property of our existence. Anything that tries to avoid that (as opposed to working with and within those limitations) is skewed away from understanding the system it is trying to describe.
(last time I said this it came across as an argument for giving up any pretence at formality and just using informal language for everything  which, as noted, was annoying given that I have attacked axiomatic mathematics for relying on informal language. That isn't my intention. Rather: there are areas of doubt and uncertainty that cannot be overcome under any circumstances. Yet we can still build precise and reproducible computers. There is a constructive way to work within those constraints.)
Cauchy:
I don't think I can come close to answering your questions in a single post. And I don't think my answer is going to be in the form you are expecting (as implied by the way you are asking the questions).
But you are asking the right questions.
1. We know that we cannot specify any definition in an absolute sense (or even know what 'absolute' means in an absolute sense).
As much as the first refrain of mathematics is "define your terms", noone has ever worked out how to do that in an absolute sense  and lots of people have tried really hard  because it would make things (including mathematics) very much easier.
On the other hand, we are capable of understanding one another to some degree at least some of the time.
On the third hand, the specific bits of interest (to me) are the raggedy edge  those bits where a general consensus is difficult to obtain  where we are talking about things far outside normal human experience and thus with little direct human experience to support an intuitive understanding of what a given word means or implies.
What does it mean for two systems to be "mostly separate, but not quite entirely so"?
A huge amount of what I want to describe depends on exactly what difference/distinction mean/are/imply.
I assume everybody has some understanding of the difference between iOS and Android. Or between their left hand and right hand.
The laws of thought specify a set of assumptions about identity and distinction. Basically: A thing is either itself or not itself. A thing is never slightly itself  because whatonearth can you do with "A thing is sortof, kindof, a little bit itself but, youknow... not entirely... it is also not itself... youknow... a bit....".
Except that I'm pretty much arguing the last bit. A thing  anything and everything  is entirely and solely specified through its relationships with other things.
In practice, this shouldn't be too much of a shock. We are already familiar with one word being specified by reference to other words. And A ⊃ B is a standard part of logic. The idea of the context defining aspects of the content is familiar.
The assumption I'd really like to cut off at the knees is that there exists anything that has meaningful properties (including, for example, existence) independent of anything/everything else.
Not the laws of thought
The laws of thought are an underlying assumption of mathematics. And I think those assumptions are wrong (or, if you prefer  I think rejecting those assumptions in favour of... something else... can lead us to a very fruitful line of enquiry).
If we continue in a conversation in which you assume the laws of thought (as any good mathematician should) and I don't  we are going to go nowhere fast.
So, before I continue trying to answer your questions... I have to ask whether you think it is possible to lay aside the Laws of Thought (even if only provisionally)?
Edit: Edit: rmsgrey snuck in a post  conventionally, !x is everything except x. While I agree that those alternative interpretations can be made  if you default to assuming the most bizarre interpretations over the well established default interpretation then the other person is always going to appear a fool.
Of course  I am now explicitly arguing against 'identity' as specified by the laws of thought...
Edit, edit, edit: When I am setting out to use terms in an odd fashion I try to flag that. When I am talking about axiomatic mathematics I try to use terms in the same way as a 'normal' mathematician would use those terms with respect to axiomatic mathematics  I'm not just throwing in my meanings willy nilly and expecting you to read my mind  honest...
(not criticising anyone else  just that Cauchy is really, really good  not just with me  he seems to have a real talent for cutting through confusion in all the threads I've seen him in).
Moiraemachy & elasto: Yes, I strongly agree with the majority of what you say. And if that was the end of the story I would shutup and accept axiomatic mathematics, warts, limitations and even contradictions, as the best we have.
However, I think there is an alternative that avoids some of the problems of axiomatic mathematics.
I'd like to describe that alternative.
The trouble is that I think there is a similar thing to Euclidean Geometry before it was realised that the fifth postulate was distinct from the previous four. Or how General Relativity is sufficiently different from Newtonian Mechanics that there isn't a direct transform between the two.
As such, when I try to describe the alternative  it looks wrong from the perspective of axiomatic thought. I'm not attacking axiomatic mathematics because I have taken a dislike to it  I'm trying to undermine some of the assumptions behind axiomatic mathematics because they appear to be getting in the way of seeing what an alternative to axiomatic mathematics might be.
For example, axiomatic mathematics tries to avoid the inherent fuzziness and uncertainty in human communication. But that fuzziness and uncertainty is a fundamental property of our existence. Anything that tries to avoid that (as opposed to working with and within those limitations) is skewed away from understanding the system it is trying to describe.
(last time I said this it came across as an argument for giving up any pretence at formality and just using informal language for everything  which, as noted, was annoying given that I have attacked axiomatic mathematics for relying on informal language. That isn't my intention. Rather: there are areas of doubt and uncertainty that cannot be overcome under any circumstances. Yet we can still build precise and reproducible computers. There is a constructive way to work within those constraints.)
Cauchy:
I don't think I can come close to answering your questions in a single post. And I don't think my answer is going to be in the form you are expecting (as implied by the way you are asking the questions).
But you are asking the right questions.
1. We know that we cannot specify any definition in an absolute sense (or even know what 'absolute' means in an absolute sense).
As much as the first refrain of mathematics is "define your terms", noone has ever worked out how to do that in an absolute sense  and lots of people have tried really hard  because it would make things (including mathematics) very much easier.
On the other hand, we are capable of understanding one another to some degree at least some of the time.
On the third hand, the specific bits of interest (to me) are the raggedy edge  those bits where a general consensus is difficult to obtain  where we are talking about things far outside normal human experience and thus with little direct human experience to support an intuitive understanding of what a given word means or implies.
What does it mean for two systems to be "mostly separate, but not quite entirely so"?
A huge amount of what I want to describe depends on exactly what difference/distinction mean/are/imply.
I assume everybody has some understanding of the difference between iOS and Android. Or between their left hand and right hand.
The laws of thought specify a set of assumptions about identity and distinction. Basically: A thing is either itself or not itself. A thing is never slightly itself  because whatonearth can you do with "A thing is sortof, kindof, a little bit itself but, youknow... not entirely... it is also not itself... youknow... a bit....".
Except that I'm pretty much arguing the last bit. A thing  anything and everything  is entirely and solely specified through its relationships with other things.
In practice, this shouldn't be too much of a shock. We are already familiar with one word being specified by reference to other words. And A ⊃ B is a standard part of logic. The idea of the context defining aspects of the content is familiar.
The assumption I'd really like to cut off at the knees is that there exists anything that has meaningful properties (including, for example, existence) independent of anything/everything else.
Not the laws of thought
The laws of thought are an underlying assumption of mathematics. And I think those assumptions are wrong (or, if you prefer  I think rejecting those assumptions in favour of... something else... can lead us to a very fruitful line of enquiry).
If we continue in a conversation in which you assume the laws of thought (as any good mathematician should) and I don't  we are going to go nowhere fast.
So, before I continue trying to answer your questions... I have to ask whether you think it is possible to lay aside the Laws of Thought (even if only provisionally)?
Edit: Edit: rmsgrey snuck in a post  conventionally, !x is everything except x. While I agree that those alternative interpretations can be made  if you default to assuming the most bizarre interpretations over the well established default interpretation then the other person is always going to appear a fool.
Of course  I am now explicitly arguing against 'identity' as specified by the laws of thought...
Edit, edit, edit: When I am setting out to use terms in an odd fashion I try to flag that. When I am talking about axiomatic mathematics I try to use terms in the same way as a 'normal' mathematician would use those terms with respect to axiomatic mathematics  I'm not just throwing in my meanings willy nilly and expecting you to read my mind  honest...
Last edited by Treatid on Sat Aug 27, 2016 1:01 am UTC, edited 1 time in total.
Re: Misunderstanding basic math concepts, help please?
Treatid wrote:Edit: Edit: rmsgrey snuck in a post  conventionally, !x is everything except x. While I agree that those alternative interpretations can be made  if you default to assuming the most bizarre interpretations over the well established default interpretation then the other person is always going to appear a fool.
Of course  I am now explicitly arguing against 'identity' as specified by the laws of thought...
Edit, edit, edit: When I am setting out to use terms in an odd fashion I try to flag that. When I am talking about axiomatic mathematics I try to use terms in the same way as a 'normal' mathematician would use those terms with respect to axiomatic mathematics  I'm not just throwing in my meanings willy nilly and expecting you to read my mind  honestly...
The thing is, !x being everything but x only makes sense in contexts where you have an "everything" to take x away from. And even if you don't have a solid grasp on "everything", the one thing that's clear about "what's left when you remove x" is that it has an xshaped hole in it  under the standard meaning of !x, there is no overlap between x and !x  if you take x away and still have some of x left, then you haven't taken x away; if you take x away and take some of !x with it, then you haven't left !x behind. Though saying that you have some x in your !x (or vice versa) also means that you're equivocating on what those terms refer to  the x that overlaps !x is not the x that defines !x be not overlapping it.
If you are using !x in the conventional sense you claim, then what you're saying about it is inconsistent; if you're saying something sensible, then you aren't using !x with the conventional meaning.
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