The most basic things are called 'fundamental'. The most basic laws are called the 'fundamental laws of physics'. The concept of 'fundamental' is of utmost importance in science. Science is seeking to explain the most fundamental reality. Science is seeking to explain everything in terms of one (ideally) fundamental theory. This "theory of everything" will be the fundamental law of physics, in the sense that all other laws can be derived from it, but it cannot be explained in terms of anything simpler.

The most basic particles, 'fundamental particles', are those that can combine to make everything else that is more 'complex'. These fundamental particles have intrinsic properties like mass. The more mass something has, the more it weighs. Every single electron in the universe has the exact same

The term 'constant' is used in physics to refer to a particular number that doesn't change, and tells us how big something is. It could be how heavy an electron is, how fast light moves, how strong gravity is, etc. All these things are finite quantities, which have particular, unchanging values that we only know through measurements and observations. These quantities are called constants.

__amount__of mass. We can quantify the amount of mass in an electron by comparing it to any proton. Every proton is always 1,836.15267245 times more massive than any electron. It is constantly that amount. Hence, we call the mass of an electron a 'constant.'The term 'constant' is used in physics to refer to a particular number that doesn't change, and tells us how big something is. It could be how heavy an electron is, how fast light moves, how strong gravity is, etc. All these things are finite quantities, which have particular, unchanging values that we only know through measurements and observations. These quantities are called constants.

How can science explain the value of the above mentioned constant in terms of something more fundamental? What determines this number? Why isn't it 2000 or 7.6453 or .000001? Why aren't electrons more massive than protons? Can science go any further? How do you explain

Richard Feynman expresses this difficulty in his book QED (page 129), with regard to one of these constants, the fine structure constant (which he refers to as the coupling constant. Don't get scared if you don't understand what the fine structure constant is. It's not essential to the proof. Think about the mass of the electron if it is easier to relate to.)

In our current conception of the fundamental laws of physics, there are 25 or so physical constants (specific quantities like the mass or charge of an electron), some of which are dimensionless physical constants (a pure number with no units. This is not as abstract a concept as it sounds. It basically just means a ratio between two things with similar units.) One of these dimensionless constants is 0.08542455, which characterizes the strength of the electromagnetic force and is directly related to the charge of an electron. (The bigger the number, the stronger the repulsive force between two electrons would have been.) The essential mystery is not tied to the fine structure constant in particular. It is just one of 25 examples. When Feynman wrote this in 1985, all these constants were shrouded in this tremendous mystery. What sense is there to specific

In order to understand Feynman's question, you have to realize what he is assuming. He is assuming that a number cannot be fundamental. This is because it makes very little sense to say that the most basic existences in reality are 25 arbitrary numbers. What Feynman is asking is that if these numbers are not fundamental, how can science possibly explain these constants in terms of something more fundamental?

An appreciation of this problem is necessary before we can move forward in the story. Specific fundamental numerical values seem to defy any possible form of explanation. It doesn't seem reasonable to believe that any qualitative physical theory will ever spit out a number like 137.03597 (and some of the other numbers are even worse). They seem totally arbitrary. (It would be a different story if the numbers we were trying to produce were 1, 3, or the square root of 2 pi; if it were numbers like these, maybe we could stand a chance at deriving them from some qualitative concept. For instance, if it involved pi, we would look for a qualitative law involving circles...) This was one of the biggest difficulties in modern physics. We had absolutely no understanding about these fundamental constants, yet they were essential parts of our equations.

Two solutions were proposed (and still are by a minority of scientists) to try to explain where these arbitrary numbers came from. The first theory simply stated that these 25 numbers were Necessary Existences (this is the theory Feynman is implicitly rejecting). Needless to say, this did not satisfy most physicists. While it is obvious that you will ultimately arrive at an idea which is irreducible and not explainable in terms of simpler concepts, it is one thing when your axiomatic ideas are nice theories such as general relativity and quantum mechanics (or maybe a grand unified theory if you prefer one eternal existence); it is altogether a different thing to have a pantheon filled by general relativity, quantum mechanics, and 25 arbitrary numbers, all necessarily coexisting.

A second theory speculated that perhaps these 25 numbers were necessary results of some qualitative Master Mathematical Equation that had yet to be discovered. This too did not satisfy most physicists as it does not seem plausible that any qualitative law would naturally generate the specificity of numbers required by observation. There was a general state of discontent with these forced explanations as they did not provide very much understanding or insight into the values of the constants.

*a number?*Richard Feynman expresses this difficulty in his book QED (page 129), with regard to one of these constants, the fine structure constant (which he refers to as the coupling constant. Don't get scared if you don't understand what the fine structure constant is. It's not essential to the proof. Think about the mass of the electron if it is easier to relate to.)

"There is a most profound and beautiful question associated with the observed coupling constant...It is a simple number that has been experimentally determined to be close to 0.08542455. (My physicist friends won't recognize this number, because they like to remember it as the inverse of its square: about 137.03597 with about an uncertainty of about 2 in the last decimal place. It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it.) Immediately you would like to know where this number for a coupling comes from: is it related to pi or perhaps to the base of natural logarithms? Nobody knows. It's one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man. You might say the "hand of God" wrote that number, and "we don't know how He pushed his pencil." We know what kind of a dance to do experimentally to measure this number very accurately, but we don't know what kind of dance to do on the computer to make this number come out, without putting it in secretly!"What was the mystery that all good theoretical physicists worried about for 50 years?

In our current conception of the fundamental laws of physics, there are 25 or so physical constants (specific quantities like the mass or charge of an electron), some of which are dimensionless physical constants (a pure number with no units. This is not as abstract a concept as it sounds. It basically just means a ratio between two things with similar units.) One of these dimensionless constants is 0.08542455, which characterizes the strength of the electromagnetic force and is directly related to the charge of an electron. (The bigger the number, the stronger the repulsive force between two electrons would have been.) The essential mystery is not tied to the fine structure constant in particular. It is just one of 25 examples. When Feynman wrote this in 1985, all these constants were shrouded in this tremendous mystery. What sense is there to specific

*numbers*being fundamental?An appreciation of this problem is necessary before we can move forward in the story. Specific fundamental numerical values seem to defy any possible form of explanation. It doesn't seem reasonable to believe that any qualitative physical theory will ever spit out a number like 137.03597 (and some of the other numbers are even worse). They seem totally arbitrary. (It would be a different story if the numbers we were trying to produce were 1, 3, or the square root of 2 pi; if it were numbers like these, maybe we could stand a chance at deriving them from some qualitative concept. For instance, if it involved pi, we would look for a qualitative law involving circles...) This was one of the biggest difficulties in modern physics. We had absolutely no understanding about these fundamental constants, yet they were essential parts of our equations.

Two solutions were proposed (and still are by a minority of scientists) to try to explain where these arbitrary numbers came from. The first theory simply stated that these 25 numbers were Necessary Existences (this is the theory Feynman is implicitly rejecting). Needless to say, this did not satisfy most physicists. While it is obvious that you will ultimately arrive at an idea which is irreducible and not explainable in terms of simpler concepts, it is one thing when your axiomatic ideas are nice theories such as general relativity and quantum mechanics (or maybe a grand unified theory if you prefer one eternal existence); it is altogether a different thing to have a pantheon filled by general relativity, quantum mechanics, and 25 arbitrary numbers, all necessarily coexisting.

A second theory speculated that perhaps these 25 numbers were necessary results of some qualitative Master Mathematical Equation that had yet to be discovered. This too did not satisfy most physicists as it does not seem plausible that any qualitative law would naturally generate the specificity of numbers required by observation. There was a general state of discontent with these forced explanations as they did not provide very much understanding or insight into the values of the constants.

What was the mystery that all good theoretical physicists worried about for 50 years?

Feynman's mystery is an abstract point. (Notice that we haven't mentioned anything about probability or fine tuning.)

Feynman's mystery is an abstract point. (Notice that we haven't mentioned anything about probability or fine tuning.)

How could arbitrary numbers be a fundamental part of reality? And if they weren't fundamental, what could possibly have caused the constants?

There had to be **some** numbers, no? Didn't Maimonides state (about the numbers of sacrifices) that if a different number was chosen, you'd be asking the same question - meaning, no number is special, there is nothing to discuss. As a matter of fact, my assumption is the opposite of yours - if the number was 2, 3, or square root of PI, I'd take a guess that the bottom-level theory has not been reached.

ReplyDeleteAs a matter of fact per Chaitin (maybe an cousin?) astronomically unimaginably humongous majority of numbers are totally random. Given this, plus the fact that there had to be some number, you should not at all be surprised by finding a random-looking constant. So I do not get what the excitement is about.

Dr_Manhattan

what would you respond if someone were to ask you where the number 7 for the sacrifices came from? What determined it?

ReplyDeletesurely you would reply that God decreed it to be so as He is the one who commanded the sacrifices themselves.

that answer is not going to fly at this stage of the game. That is not the most compelling proof of God, to say the least.

Rambam's answer is NOT that "well, God picked the number!" but that the entire question is pointless. When some relationship in the Universe is quantifiable, and there is some quantity, there is no point asking why this quantity is unique, because had it been any other quantity you'd be asking the same question. Mathematically a near-perfectly random (uncompressible) number is what you would expect, and what you in fact get.

ReplyDelete> that answer

I didn't actually answer anything, I defy the premise of the question

>"fly at this stage"

ahem, there is not much of a stage to fly over frankly...

And yes, I agree that 137.03597 is not the most compelling proof of God. I'll stay tuned though.

Dr_Manhattan

The Rambam in his philosophical book is addressing why God picked 7. In so far as there is no discernible difference between 6,7, or 8, it is unreasonable to seek a purpose in 7 specifically being selected over 6 or 8. (It is reasonable to conclude that somewhere in the range of 7 had to be selected as 1000 sacrifices every holiday would be practically impossible, in addition to causing severe economic shocks to Jewish society.)

DeleteThe Rambam is not addressing what caused 7. That is because it is known that the cause of 7 is the decree from God in the Torah. This is called under the system of halacha (Jewish law) a gzeirus hakasuv (a decree from a biblical verse).

Scientists, are seeking a similar type of cause for the constants. What determined the constants? Have we reached the end of the line in terms of understanding them? Do we just have to say "as far as we know, they are that way because they are that way and there's nothing more we know?"

Dr_Manhattan,

DeleteAre you suggesting the necessary existences approach? If not, please differentiate.

I guess necessary existences, lowercased (Uppercasing Them would sound like they have wings and sing hymns, and IMHO that theory has ran its course)

DeleteDr_Manhattan

"[The constants] seem totally arbitrary. (It would be a different story if the numbers we were trying to produce were 1, 3, or the square root of 2 pi; if it were numbers like these, maybe we could stand a chance.)"

ReplyDeleteDoes this mean to suggest that a "Master Mathematical Equation Theory" would be plausible if the constants were in some sense "significant", i.e., have a prior usage or application in physics? I argue that this would still fall prey to the "qualitative doesn't explain quantitative" breakdown; for if not, then one might argue that these constants have some "significance" (in a similar to sense to, say, pi) that simply hasn't been discovered yet by physicists.

If the constant was pi (the mathematical constant that expresses the ratio between the circumference of a circle with its diameter in Euclidean geometry), there would be room for hope, and speculate that some qualitative argument (i.e. the isotropic nature space or the symmetry of physical laws under rotation) would somehow yield that constant.

DeleteThe essential point is that as far as anyone can tell, these physical constants have no relationship to the mathematical constants (pi, the natural logarithm, the golden ratio). It's a largely academic question as to what scientists would have said if they did.

Many papers were written in the past trying to reduce the fine structure constant to the mathematical constants and they all failed. Scientists have long abandoned that approach (especially with the recognition of the cosmological constant which has 120 decimal places).

Hard to follow...can you take the writing down a notch so anyone unaffiliated can grasp all your points?

ReplyDeleteWhat is this number: 0.08542455

A weight? A distance?

What is a "fundamental"?

Feynman's words are cryptic to me. Are you saying this number is found throughout the universe? If so, where....please provide a few examples.

Thank you.

the number 0.08542455 is a measure of the strength of the electromagnetic force, which is one of the 4 fundamental forces in nature.

DeleteIt has no units. It is just a pure number. (This is possible because it is a ratio between the electron's charge and the speed of light and Plank's constant). Another example of a dimensionless ratio is the ratio of the mass of an electron and a proton which is 1836. It is simply a number with no units.

'Fundamental' means that it's not derivable from something else more basic than it.

This number is found throughout the universe, in so far as electrons and protons are everywhere. Wherever there are atoms, this number is relevant.

Think about it as a number that you multiply the electromagnetic force by. If it were bigger, the force would be stronger. If it were smaller, the force would be weaker.

"The essential point is that as far as anyone can tell, these physical constants have no relationship to the mathematical constants (pi, the natural logarithm, the golden ratio). It's a largely academic question as to what scientists would have said if they did."

ReplyDeleteWhat is a mathematical constant as opposed to a physical constant?

see http://en.wikipedia.org/wiki/Mathematical_constants

Deletea physical constant is known only through observation and measurement, as opposed to a mathematical constant which is know through a priori mathematical reasoning

I noticed that you changed this post. Do you think it would be possible to notify readers of when you change posts? When I modify a post on my own blog, I either repost the entire thing or - if that would disrupt the flow - I add a line in red at the top of the post saying, "Last Updated: Wednesday, June 20th at 11:45am" or something like that.

ReplyDeletegood idea

Delete3 questions.

ReplyDeleteFirst question:

“The essential mystery is not tied to the fine structure constant in particular. It is just one of 25 examples. When Feynman wrote this in 1985, all these constants were shrouded in this tremendous mystery. What sense is there to specific numbers being fundamental?”

What do you mean by the “sense” of numbers being fundamental? Do you mean what caused the numbers to be what they are, i.e. how were they selected or what selected them? Or are you saying there is a problem with numbers (as opposed to qualities like gravity) being fundamental? And if the latter, why is that a problem? Also is it a problem on its own or only because of the fine tuning?

Second question:

“While it is obvious that you will ultimately arrive at an idea which is irreducible and not explainable in terms of simpler concepts, it is one thing when your axiomatic ideas are general relativity and quantum mechanics (or maybe a grand unified theory if you prefer one eternal existence); it is altogether a different thing to have a pantheon filled by general relativity, quantum mechanics, and 25 arbitrary numbers, all necessarily coexisting.”

Why is this a problem? Why would just having axioms of relativity and quantum (a pantheon of two) be superior? What exactly is it “altogether different” about having 25 constants as a part of your fundamental laws?

Third question:

“They seem totally arbitrary. (It would be a different story if the numbers we were trying to produce were 1, 3, or the square root of 2 pi; if it were numbers like these, maybe we could stand a chance.)”

Please explain how a constant of 1 or 3 would change the problem, or allow us to “stand a chance.” What relevancy does this have? E.g if hypothetically 3 was the number required for fine tuning how would the question of be lessened?

Thanks!

Excellent questions. Your questions are all very much related.

DeleteFirstly, 'fundamental' is being used in the sense of not having a cause. Not being explainable in terms of anything simpler (more fundamental).

It is critical to realize that at this stage (post 2-the mystery) the question of "How could arbitrary numbers be a fundamental part of reality?" has nothing to do whatsoever with fine tuning. Fine tuning is not introduced until the next post. The fact of fine tuning is what delivers the death blow to this theory.

The problem Feynman is expressing over 25 arbitrary number like 0.08542455

being the most fundamental parts of reality is because it is an incredibly ugly theory. It does not sit well with any good theoretical physicist that ultimate reality is 0.08542455.

The difference between 0.08542455, and general relativity and quantum mechanics can best be appreciated by someone who understands the breathtaking mathematical beauty, symmetry, and simplicity expressed in those theories. The greater your knowledge, the greater a sense of wonder you will have when you studies those theories. (A good way to get a sense of this is to read quotes from great theoretical physicists like Einstein and Feynman about the beauty of the natural law.)

General Relativity and Quantum are two of the most elegant theories the human mind has ever beheld. While mathematical beauty and simplicity do not prove that a theory is true, it is strong criterion for not accepting 25 arbitrary numbers as ultimate reality, and most people can appreciate this point even without having understood Relativity and Quantum.

Lastly, if these numbers were related to mathematical constants we might hope to explain them in terms of beautiful, simple qualitative laws. For example, if the number were pi, maybe some qualitative argument could be made from geometry based on the symmetry of a circle. (It's hard for us to make up plausible, beautiful theories for facts that don't exist. we hope that example helps you grasp the concept.)

The punch line is that if they were mathematical constants, we could hope that there really is one unique, beautiful, simple mathematical theory that determines all the constants and that numbers like 0.08542455, wouldn't be part of the ultimate reality.

ReplyDelete"it does not seem plausible that any qualitative law would naturally generate the specificity of numbers required by observation."Why not?

See our response to Rinde above. Does it sufficiently answer your question?

DeleteFirst of all thank you for taking the time to write these posts, I am enjoying them but I have a question as I can't find myself agreeing with one sentence:

ReplyDeleteI agree with Agur, in that I don't understand " that it doesn't seem plausible that any qualilative law would naturally generate the specificity of numbers required by observation."

The number in Feynman's quote (and I'm referring to the inverse of 137.xxxx), to which he is referring to as the "fine-structure constant", is not as precise as the number believed to be the fine-structure constant today. As you have mentioned, these numbers are experimentally derived and are only as precise as the current available methods of measuring them. Since Richard Feynman's quote, accoring to wikipedia, since they give me more decimal places, I'm assuming that there has been an increase in the precision of what the fine structure constant.

I have no reason to believe that the current precision is going to be the end of that constant's decimal places as well, just that we don't know yet what those additional millionths, zillionths, etc... are.

How ugly pi looks truncated: 3.14. Its beauty rests in the never repeating, patternless afterworld. Who is to say that this fine structure constant doesn't have such a pattern if only we could calibrate our machines well enough?

That being said, were the fine structure constant to have some sort of transcendental (or perhaps even irrational or rational pattern), would that increase in your mind the possibility that it could be derived from some natural law?

I guess I can see where it might not be "plausible" or "likely" that these numbers are generated from an equation. However I cannot accept that it is not possible. What do you think?

We're glad you're enjoying the posts. So are we.

DeleteIf the fine structure constant had any way of being simply expressed through mathematical constants like pi, it would greatly increase the plausibility of some qualitative explanation determining it. (Meaning, perhaps the fine structure constant is related to the geometry of a circle.)

However, if it was just an arbitrary repeating sequence (a pattern), we don't think it would help very much with the problem of how a qualitative equation would produce that specific pattern, without relating it to the mathematical constants.

A repeating sequence for the fine structure constant would help with the second problem of the qualitative equation having to determine the constant to the very last decimal place. As of today, we know the fine structure constant to eleven decimal places and it isn't repeating. We know the cosmological constant to over 120 decimal places and it isn't repeating.

It is not logically impossible there exists such a qualitative equation of a physical law that determines all the constants perfectly. However, all good theoretical physicists realize that this is a highly implausible theory, even though they harbor the wishful fantasy that it does exist