baric,

"For example, if we do encounter alien intelligence some day, what percentage of them will have developed math vs. music? It would seem to be 100% (math) vs. some number less than 100% (for music)."

My opinion is that if they have alien maths they will probably have alien music and vice versa. They both evolved at the same time (more or less) in the human developmental vector.

"Maths are "emergent" precisely because they correspond to the fundamental nature of the universe. To compare them to transient phenomena like tennis or classical music is unreasonable."

No - human maths emerged through the evolution of homo sapiens. Maths is a framework for us to understand and manipulate our physical environment to a reasonably accurate (though not perfect) degree. Just because our maths is a very good approximation in no way means that the ultimate laws on which our universe is based are "mathematical" or "computational".

For instance, You may be able to make a really good counterfeit of a genuine designer label product, but the production process of the two items could have been radically different. If we follow your argument then one would always assume that the fake was made exactly as was the genuine. Thats a false assumption.

That only strengthens the point. Feynman had a more limited form of synethesia, but yet was a remarkable mathematician.

I seriously doubt that article addresses what I am talking about. I am less interested in how "increased structural connectivity" causes this neurological condition as I am why this visualization of numbers translates into remarkable computational ability. That is what we cannot explain.

They see numbers as we see material objects, and this sight provides measurable and repeatable abilities that are beyond the ability of people who don't have this ability.

I am a believer that our minds, engineered for biological survival and reproduction, are poorly-suited to understanding the inner workings of the universe. But occasionally, we have savants who by fortune get a differently-wired brain that seems less suited for survival and more suited for math. History is sprinkled with the works of rare math prodigies who thrived because society had progressed to a point where non-savants like us developed an appreciation for their gifts.

There is something fundamental about the nature of math that you and I are completely incapable of comprehending. It's not an educational issue, it's a flaw in how our brain is wired. It certainly irks me to some degree, but just because I cannot see what they see does not invalidate it.

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"Barbarism is the natural state of mankind. Civilization is unnatural. It is a whim of circumstance. And barbarism must always ultimately triumph" -- Conan

You can't comprehend its fundamental nature, but somehow know it's true? That's not good enough for me.

And your point is that just because one can't see your theory of the fundamental nature of math, it does not invalidate it? It sounds like you're saying "Prove me wrong." No, prove your assertion right. You're the one making the claim.

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Plan as though you will live forever, but live as though you may die tomorrow.

I'm not sure I see it that way. I am reacting to the claim that "math is emergent" by showing someone who uses a technique for math that was not only self-taught, but is completely alien and unusable by anyone else. To me, that makes a serious argument against the "math is emergent" claim.

Where is the evidence that math is emergent?

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"Barbarism is the natural state of mankind. Civilization is unnatural. It is a whim of circumstance. And barbarism must always ultimately triumph" -- Conan

Well, "structural connectivity" is what enables human thought. If you were learning arithmetic in 1st or 2nd grade and you noticed all these colors and patterns that were above and beyond a normal student's experience with arithmetic, you might get pretty good at it. If you continued to develop the ability, adding in some simple numerical analysis, you might become quite exceptional.

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Everyone is entitled to his own opinion, but not his own facts.

This is a very short theory yet likely and enjoyable
If a creation that holds an "AI with a vastly higher IQ than human", will start searching after the creation of the great "un-computable", holding the capacity or power to learn, beyond or under a specific IQ level, will possess a different perspective about his and the collective existence, and about its origin;
Just like a squirrel won't wonder about the BH more than a human, a highly intelligent AI will probably possess a different, even an opposing view of the collective.

This question is pretty subjective, but for this matter, if the AI has a consciousness and the capacity to learn, it might just define ideals such as infinities and time, in a way, whether contradictory to the human perspective and whether not, so that it would fit into his view of the collective, of the individual within the collective and of the "structural connectivity by his defining .

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Music only gets better with time

BTW, there are 1.5 billion synapses in 1 cubic millimeter of the human brain....

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Everyone is entitled to his own opinion, but not his own facts.

I find it difficult to ascribe this ability simply to increased computational power. Think about the known ways to factor a large number: iterating through a known list of primes and doing repetitive divisions.

If someone were to ask me to factor a 4-digit number, I could do it fairly easily within a minute or two, mostly as the result of a childhood fascination with primes. However, it is a computational approach that increases linearly with the size of the number.

But this particular person can factor very large numbers (10+ digits iirc), and he describes doing the process in a visual manner -- one that frankly makes no sense to me. It's not a matter of just doing it more quickly because he has greater "structural connectivity" in his brain. He sees numbers differently and is able to manipulate them in novel ways.

All I'm saying is that, if I were a platonist, I would find this guy to be convincing evidence of that philosophical framework. I only brought him up in response to the presumptive declaration that math is emergent.

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"Barbarism is the natural state of mankind. Civilization is unnatural. It is a whim of circumstance. And barbarism must always ultimately triumph" -- Conan

Well, I don't know what process he goes through either. But I do know that when one becomes very familiar with numbers and some of the patterns they fall into, there are "tricks." Can you multiply 63 x 67 in half a second? Yes, if you know the trick* -- when the tens digit is the same and the ones digits add up to 10.

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* Multiply the tens digit by one more than the tens digit. In this case, 6x(6+1)=6x7=42. Now just multiply the ones digits together. 3x7=21. Now just "cramming" the results together gives you 4221. Amaze your friends and neighbors!

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Everyone is entitled to his own opinion, but not his own facts.

It's not really linear, is it? You only have to check for factors up to the square root of the number. So the computation time only increases as the inverse square of the increase in the number...

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Everyone is entitled to his own opinion, but not his own facts.

Yes, those are easy tricks but they do impress people. Also don't forget about two multiplicands that center around a known square (72x88=6400-64=6336). There is no known trick to factoring primes, however, which is why that particular example astonishes me.

It increases in time by the root only if you can mentally divide by 73 as easily as you can divide by 7! I can't, so I estimated back up to a linear rate.

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"Barbarism is the natural state of mankind. Civilization is unnatural. It is a whim of circumstance. And barbarism must always ultimately triumph" -- Conan

baric,

"I'm not sure I see it that way. I am reacting to the claim that "math is emergent" by showing someone who uses a technique for math that was not only self-taught, but is completely alien and unusable by anyone else. To me, that makes a serious argument against the "math is emergent" claim.

Where is the evidence that math is emergent?"

The evidence is all around you in that all human subjects are emergent, whether maths, english literature or even Tennis. Did those subjects as we understand them exist before homo sapiens evolved? The answer is no, or at least not to our knowledge. We invented those formulations over thousands of years of observation and trial and error. We can prove that.

You have zero proof to demonstrate that these formulations existed before we arrived through evolution.

I reject your claim that math is something as obviously subjective like english literature or tennis. How many different possible ways are there to formulate a counting system? Is there a counting system that has different prime numbers?

Was the speed of light limited to c before Einstein formulated special relativity? If alien civilizations have a different way of thinking, is it possible for them to formulate a theory that allows travel faster than c? Or is c a constant, regardless of whether any species has developed an intelligent sufficient to comprehend that?

I would counter that it's more likely that mathematical constructs are universal, and that we are simply discovering them as needed to solve problems, much as we discover other truths about the universe.

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"Barbarism is the natural state of mankind. Civilization is unnatural. It is a whim of circumstance. And barbarism must always ultimately triumph" -- Conan

Mathematics covers a large domain space. It was developed from counting objects and leading to simple arithmetic (for which you need to have language, otherwise you cannot communicate precise concepts). That much was derived from the physical environment. However, there are fields in mathematics that have developed through agreed upon conventions which have nothing to do with the physical world. So while an alien race is likely to have 'discovered' mathematics and is likely to overlap our formulations to some extent, it's also likely that they will have esoteric formulations and conventions that would never have occurred to us. So in that sense, mathematics can be just as subjective as a natural language like English.

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Conscious reasoning is an attempt to justify the choice after it has been made.

Mathematics cannot be subjective, the path of the learning, the language or the way its being taught, the time it takes to teach a specific item and countless more variables are entirely subjective, but the knowledge its self, is not.

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Music only gets better with time

baric,

"I reject your claim that math is something as obviously subjective like english literature or tennis"

Well you're deluding yourself. Maths is a human formulation just like english lit. Are you seriously suggesting that our maths holds some special virtue of universal objectvity? If so, then you need to prove that assertion because its a biggey.

"How many different possible ways are there to formulate a counting system? Is there a counting system that has different prime numbers?"

We've had different counting systems throughout our own history. And are you seriously suggesting that if we meet intelligent aliens they will have the same counting systems as ours? Thats a hell of an assumption. You seem to have come to the conclusion that the great human race sits on top of some apex of intelligence about nature and the universe. Sorry I dont share your sense of omniscience.

"Was the speed of light limited to c before Einstein formulated special relativity?"

You've confused the point. I'm talking formulations. You are talking about observables.

However, nobody knows for certain whether and which (and by how much) constants - such as hubble - have changed through the course of the universe and there are many experiments underway to answer that question. There are indications that the speed of light may have been slower just 2 billion years ago right here on earth. But neither you nor i can answer that question until the real scientists prove it one way or another.

As for aliens measuring the speed of light. I assume they would do so with their own system whatever it may be. But i don't accept its going to be just like ours.

"I would counter that it's more likely that mathematical constructs are universal, and that we are simply discovering them as needed to solve problems, much as we discover other truths about the universe."

Well I await proof of that. It would be wonderful to think that consciousness and awareness wherever its found in the universe has discovered the exact same way to describe and manipulate the universal environment. But i fear its wishful thinking.

What do you mean by 'knowledge'? A mathematical proof? Proofs are derived from a set of axioms - whenever you start from the same axioms you'll get the same result for that proof, true. However, the choice of axioms is somewhat subjective as there is nothing to say that axiom set A is any better than axiom set B.

And what of mathematicians' appreciation for mathematical 'beauty' - surely a subjective bias? Why should one proof be more desirable than another when both are logically sound?

I won't even mention Gödel's incompleteness theorems...

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Conscious reasoning is an attempt to justify the choice after it has been made.

New axiom suggestion: "All true statements that can't be proven by the other axioms are true." This solves the Gödel incompleteness problem.

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Life is like a box of chocolates. All of your choices are bad for you.

Please enumerate. (pun intended)

Yes, I am suggesting that. SETI researchers are assuming that as well.

I have never suggested that we are the apex of intelligence. In fact, this discussion veered toward mathematics precisely because I was discussing how limited our intelligence is.

You, on the other hand, have declared math as clearly 'emergent' without any evidence beyond that it's like tennis or literature, and that I'm 'delusional' for disputing that connection. Your historical evidence for emergence can easily be presented instead as evidence of discovery.

I am sure that I am not the only person on this forum who would like you stop using ad hominems to buttress your point and present these "alternative systems of counting" that have different prime numbers.

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"Barbarism is the natural state of mankind. Civilization is unnatural. It is a whim of circumstance. And barbarism must always ultimately triumph" -- Conan

Firstly, I mentioned the mathematics as a whole.

Secondly; In this case, the subjective mathematics for example of a 6 years old is indeed different from a prof's knowledge and less wide. The mathematics as a whole is not different at all, in cases of choosing different axiom systems within the whole mathematical pattern, the axioms cannot be defined by a proof, which means that in a different language, and with many different variables, the axioms of the universal mathematics will never change and they will always will lead in a way to the next universal axiom, and from here follows the whole beauty of mathematics; 1+1 will always become two.


The mathematics as a whole is not subjective, only the mathematics and the mathematical ideas of an individual are subjective.
The mathematical beauty and the appreciation of it is unfortunately not part of the collective mathematics, only of the subjective's.

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Music only gets better with time

In fact, I've never made that assertion. An excerpt from a previous post:
An example of this might be imaginary numbers (the square root of negative numbers). Before their invention, mathematics could not solve this problem because no number could be squared to a negative number, like -1. So, do imaginary numbers really exist or are they just an invented convention which has become accepted over time?

I realise that the use of imaginary and complex numbers has great utility in solving real world problems, but are they really a feature of the universe or just a handy invention? Why not come up with a similar convention for solving division by zero?

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Conscious reasoning is an attempt to justify the choice after it has been made.

baric,

"Yes, I am suggesting that. SETI researchers are assuming that as well."

And they've had plenty of success so far :-) SETI researchers have always been bound by the human perspective, as are you and I. So SETI is not a good example because so far they have found squat. While SETI have to make these assumptions because they have no other example other than ourselves; that doesnt prove or disprove whether aliens would use human maths. Thats basically what you are suggesting with zero proof.

"You, on the other hand, have declared math as clearly 'emergent' without any evidence beyond that it's like tennis or literature, and that I'm 'delusional' for disputing that connection. Your historical evidence for emergence can easily be presented instead as evidence of discovery"

Yes i absolutely stand by the assertion that human maths is an emergent property, just like every other formulation or system which has evolved or developed through the course of universal history. I dont need to prove what is widely accepted by most scientists. Emergence is widely accepted by the majority of the scientific community.

"I am sure that I am not the only person on this forum who would like you stop using ad hominems to buttress your point and present these "alternative systems of counting" that have different prime numbers."

LOL call the cavalry - it wont save you :-)

Have you even bothered to check if there were different counting systems before we settled for one with 0s? Clearly not because if you had you would find a tremendous amount of literature studying how we formulated our current counting system.

You appear to suggest that we immediately discovered our current counting system overnight or something. Thats just plain wrong. It didnt fall in our lap as if some telelogical present from an onmipotent being. It took thousands of years to formulate. Do some google searches and you'll see for yourself.

So the FACT is that we have had different counting systems in our history.

Here's how I interpreted it.

You claimed X.
I disputed X.
You claim it is "obvious" and that the burden is upon me to google up the proof.

Perhaps you are confusing the representations of numbers with the numbers themselves.

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"Barbarism is the natural state of mankind. Civilization is unnatural. It is a whim of circumstance. And barbarism must always ultimately triumph" -- Conan

That's not an XOR "or" if you're a pragmatist.

You probably know that there are a great many number spaces out there. Some have a convention for that. But which one is right depends on the theoretical construct or application. Thus, competition, and alas, no easy answer.

I'm no mathematician, but I believe Euler and others have even tried to connect -infinity to infinity as some sort of number ring. Think about f=1/x, where x becomes tiny, so f goes to infinity. Now let x=0; f is undefined, as that's what we are trying to understand. Now let x become a little smaller, ie negative; now f=-infinity. Thus it would seem to define 1/0 would be to somehow connect the idea of the infinite extremes. Note that n/x will connect the two infinites for any number n≠0. For n=0, it connects two values of zero, so it would seem like n/0=0 is a good definition.

There is a serious issue, though, if you think you could just define as you like and use standard arithmetic conventions. Consider this:
5/0=0
17/0=0
Therefore 5/0=17/0
Cancel the zero denominator on both sides
5=17
Yea! Numbers are meaningless! Boring old school's out forever!

I have claimed human maths is an emergent property which is a rather boring and mainstream statement. You have argued the following: That the underlying fundamental laws of the universe are mathematical, computational, and even more incredibly, are based on human maths! That is an extraordinary claim and requires proof.

My case against your wild conjecture (as if i even needed one) is that human maths emerged becuase of human evolution. So no human evolution= no human maths. Its that simple - you should be able to get your head around that piece of simple inane logic.

Next came your ridiculous and absolutely incorrect statement that we have always had the same counting system. Thats nonsense and just because you cant be bothered to look up the history of human maths and how we came to the current counting system is not my problem.

"Perhaps you are confusing the representations of numbers with the numbers themselves"

No you are confused in that you seem to think numbers have some independent existence outside of a numerical or counting system. They do not. Numbers are an inherent property of any counting system (that we know of).

I'd also like to add that i think its reasonable to speculate about whether the underlying and fundametal laws (outside of our reality and experience) are mathematical/computational. However its always going to be unsupportable unless we cracked some sort of code which would literally explain EVERYTHING.

What i do find interesting about modern maths in relation to how the universe really works is how it's the probability/statistical theories which appear to "work" better than the old ideal of newtonian exactitude for manipulating the physical environment. Whether macroscopic non-linearity or subatomic uncertainty the solutions for studying these systems and making predictions are always approximations.

Isnt that a more system approach? Add to that quantum entanglement which has put a huge hole in the once sacred idea of "seperability", and science is leading us into another direction, away from the old reductionist approach.

There is a difference depending on what exactly you mean, but that's a bit of an authorative and superficial statement without being more specific when speaking about such a deep and malleable concept. The representations in many ways define our understanding of the concepts. Three sticks on the ground is just a "representation" of the number three, and may become symbolic, as it has in the writing systems of many cultures, including ours. "3" is just a quick scratching of three lines, with the bottom line removed for simplification.

Actually, I did. What I got was a history of various representations for counting, from cuneiform to arabic to babylonian to roman. That's why I think you are confusing terms.

In every human representation of counting, the underlying numbers are the same. If you want to make the "wild conjecture" that this is not true, then you need to provide support for that instead of just dismissing it as a "boring and mainstream statement".

Waving your arms and making increasingly polarizing claims does not make your argument more convincing.

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"Barbarism is the natural state of mankind. Civilization is unnatural. It is a whim of circumstance. And barbarism must always ultimately triumph" -- Conan

Counting merely requires the classification of discrete objects. The classification may be subjective, but the discreteness of material is self-evident.

One of the fundamental tenets of science is that physical laws are applicable everywhere even though we may not fully understand the laws. The transition of Newtonian physics to Einstein is a perfect example of how we refine this understanding over time.

If discrete and classifiable objects exist, then it is possible for a cognizant being to count them. This counting will be the same regardless of the previous evolution of that being unless we are suggesting that objects exist only subjectively.

I think that it follows necessarily that the underlying, abstract framework required for counting must be consistent. You simply cannot have a coherent counting system, for example, where 2,3,5,7, etc are not prime.

The disagreements here may revolve around 3 separate points of contention:

1) abstractions vs. representations - when some people take about different counting systems, they may be referring to representational differences.. i.e. 1,2,3,4,5 vs. I,II,III,IV,V. When I speak of numbers, I am clearly referring to abstractions.

2) subjectivity vs. objectivity - I am working from the premise that material objects exist independently of our perception of them, and that we can reasonably confirm the actual existence of those objects through scientific study. Perhaps it is possible to formulate a universe where objects exist conditionally based on the observer (I'm not talking about quantum effects, which are predictable), but that universe is clearly not the one we are living in, so it's more of a thought experiment.

3) emergence vs. universality - the idea that numerical frameworks emerge implies that two separately evolving intelligences with different perceptions could create different numerical frameworks. I believe this is fairly absurd, considering that we deal with numbers abstractly, not perceptively. Matter operates in a similar fashion across the universe, creating identical conditions for classification and counting for intelligent beings. While they may indeed perceive things differently, they will develop the same abstractions for counting and complex maths that we do because, once you start counting, the rest of the framework necessarily follows. They will still develop zero, negative numbers and imaginary numbers. If they perceive at least two dimensions, which is reasonable, they will have geometry, trigonometry, and calculus. They may have much, much more than that based on their perceptions and intellect.

You can argue that the mathematical framework is a universal abstraction without arguing that numbers are "real" as Platonists might. I don't believe numbers are real, but I do believe that they are universal. I also do not believe that we have a strong understanding of this framework, although we scratch out a little further into it with each passing century. This is probably why rare savants can exist and demonstrate mathematical prowess that we cannot yet grasp as a species.

If someone wants to suggest that other counting systems are possible or have emerged in human history, then they are going to have to support that with a more than "it's on google, look it up". These other systems clearly haven't emerged, which is why I think that claim is based on a misunderstanding of terms in the discussion.

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"Barbarism is the natural state of mankind. Civilization is unnatural. It is a whim of circumstance. And barbarism must always ultimately triumph" -- Conan

It reminds me of an Isamovism, where a professor had explained something similar about concrete and abstract divisions of reality. He had classed math and number systems as definite ideals. Isamov walked up to speak, and the conversation went something like..
"What is this?"
"A piece of chalk."
Asimov broke it, and held up a piece.
"What is this?"
"Half a piece of chalk."
"Still looks like a piece to me."

[addition:] I actually looked up the anecdote. Sorry, my version is all wrong. But still, I'll leave the apocryphal account, because, hey, free dummy.