Did you ever hear the one about the dyslexic, agnostic, insomniac? He stayed up all night wondering if there really was a dog.
Like many people, I’m curious about the nature of reality and really do sometimes stay up all night wondering about… well, just about anything.
A while back I wrote this post about the ramifications of a comprehensible God. If God and reality are comprehensible then using reason and rationality to explore reality is a worthwhile goal. But if God and reality are not fully comprehensible, then reason and rationality will only work haphazardly, and therefore are not reliable guides.
A Slice of PI
What I find so fascinating about logic and reason are that they do work. For example, what is the value of PI?
PI is defined at the ratio of the circumference to the radius of a circle. In school, I was always taught to use 3.14 as the value for PI. But actually this is just an approximation. Supposedly, PI actually goes on infinitely past the decimal, apparently never even repeating. You can’t even turn PI into a fraction in the form of one whole number over another, so this makes it a classic so-called “irrational number.” .
But have you ever thought to ask how they know all that? How do they even know that the ratio of the circumference to the radius of a circle is approximately 3.14?
Do they take a measuring tape and measure it and find out its 3.14? I confess, I once read a book that suggested actually trying to measure the circumference and radius of a circle to calculating PI this way. I got 3.16 as my answer. Presumably the reason I didn’t get the right answer is because my measuring tape wasn’t infinitely accurate and because my circle wasn’t a perfect circle.
Actually, true circles doesn’t even exist in physical reality. But if true circles don’t exist in real life, then how the heck can we confidently say that the ratio of the circumference to the radius of something that does not exist is 3.1416…etc?
Maybe since true circles are just imaginary anyhow that’s how we came up with it? If I decide that there is something called a barf-fat and it only exists in our minds, then can’t I pretty much make up anything about it that I want? If this is true, then why can’t we just take a vote and decide to change the value of PI to 3. Now personally if PI had a value of 3 there would be many advantages over an infinitely long number that is impossible to memorize. Think about how much easier it would be to teach children about PI if its value was 3. Plus, think of all those mathematics and physics equations that use PI that would suddenly be easier to calculate. Since circles don’t really exist except as figments of our imagination, I am going to personally start writing to my congressmen today to make sure the “PI is 3” law passes the next time congress meets.
What is Real?
So just exactly how do we know what the value of PI is if we can’t measure it?
Well, here is one way you could do it. Take a look at the following picture that I took from Mathematics for the Million: How to Master The Magic of Numbers. I love this book because it teaches math from a historical perspective, showing what problems the ancients were trying to solve as they discovered various mathematical principles.
As this picture shows, we can take our ‘imaginary’ perfect circle and pretend to split it up into boxes. The boxes actually come in two sets, those that bound our circle (i.e. the circle is inside of the boxes) and those that our circle bounds (i.e. the boxes are inside the circle.)
Now imagine that each of these boxes is exactly equal in width. Note that this means that the “inner” and “outer” boxes in a quarter section are all exactly the same except for one box. In our example we have one quarter circle with 9 boxes and one with 10. But other than that extra boxes, the boxes are identical.
Now let’s assume that our circle is a unit circle, so by definition its radius is 1. It doesn’t matter what it’s “1” of. It could be 1 foot, or 1 inch, or 1 mile. It does not matter for our purposes so long as it’s “1” of some unit. Maybe think of it being “1” on one of those number lines that you used to draw on back in high school that had no units.
Now if you remember back to your old geometry days, you might remember that the area of a circle is PI*Radius^2. [1] Given that our radius is known to be 1, we know that means the area of our circle is equal to PI because 1 squared is still 1 and PI X 1 = PI.
Now if we know that the boxes are exactly equal in width and we know the radius of the circle is 1, then we already know the width of each box. From here we can use geometry to figure out the area of the boxes. [2]
The end result is that we now have the area of two sets of boxes, one that is known to be larger than the area of the circle and one that is known to be smaller than the area of the circle. For our example picture, the result would be 3.44 for the outer boxes and 2.64 for the inner boxes. So we can now be absolutely certain that PI is less than 3.44 and greater than 2.64.
Now double the boxes. Then double them again. In fact, double them as many times as you have patience for. The end result is obvious: you are now able to calculate PI to any level of precision you want. Just keep adding boxes until the upper and lower bounds match out to as many decimal places as you wish. Throw the rest away. [3]
You now know at least one way to calculate PI.
What is so fascinating about it is that you know it works. In your heart, now that you understand how it’s done, you know it’s completely reliable. Legislating what PI is now seems as silly as voting on whether or not the dinosaurs once lived on the earth.
Now how is it that a purely mental concept like a circle – and remember, it does not physically exist anywhere in universe – can have such a specific and computable characteristic like PI?
I’ll tell you why. It’s because circles really do exist. I can’t get over the significance of this. Something that only exists in our minds actually and really objectively exists and we can prove it beyond doubt. Mathematics is real.
I also can’t miss the display of beauty in math and reason demonstrated by this example. I’m left with several questions worthy of further discussion:
- Why is it that made up things can really exist objectively?
- Is Math (like circles or PI) “invented” or “discovered?”
- I mentioned that this example was “beautiful” to me. Is that a subjective “in the eye of the beholder” sort of thing? Or could beauty be objectively real too?
- What do we even mean by the word “beauty” in the first place? Why is math, nature, and my wife all beautiful? Is that one word for three things or one word for one thing?
- Do you feel God’s presence when you see math work like this? If not, do you at least feel awe for “something” when you see the above example?
Notes
[1] See Mathematics for the Million, p. 217 and p. 154
[2] Mathematics for the Million, p. 217-218 for full details
[3] Actually, this algorithm to solve for PI is what we call intractable. Though in principle we can find PI to any arbitrary decimal point, in practice once you get past a certain point the procedure becomes too computationally intensive and even a super fast computer will never complete prior to the universe collapsing into a big crunch – at least not with current technology. There are better ways to find PI that aren’t so process intensive using calculus. See this link here for details.
Wheat & Tares 
1. Why is it that made up things can really exist objectively?
I often wonder about the underlying nature of reality and the existence of platonic forms. Perhaps the scripture about all things being created spiritually first is a reference to this.
2. Is Math (like circles or PI) “invented” or “discovered?”
I am firmly in the discovered camp – the universe pushes back too often to allow math to be just an invention.
3. 4. I think that aesthetics is perhaps the murkiest of all philosophical inquiries. I took a course once on the philosophy of music, and at the end, I realized that I was totally unable to come up with a reasonable definition for music, even though I could generally recognize it when I heard it.
One can often find majority agreement on whether something is beautiful or not (with outliers, of course), and that suggests that there is some objectivity to it. However, our emotional connections add a hugely subjective component to our aesthetic appreciation. Additionally, there is the impact that the brain has on perception – one can gain and lose artistic feeling and sensitivity in surprising ways.
5. Do you feel God’s presence when you see math work like this? If not, do you at least feel awe for “something” when you see the above example?
Yes. It awes me that the universe is so rational and that we can attain understanding of it by applying reason and logic. I really see God as a scientist as much as anything else. He creates and manipulates a universe in an orderly and reasonable way to achieve a desired end. (Perhaps a scientist-engineer?)
Max Tegmark has an article that I can’t quickly reread that, as I recall, suggests that ANY coherent description of reality MUST be mathematical in nature. So maybe God is mathematical and so mathematics makes us up.
So, instead of choosing invented or discovered, perhaps in its upper reaches, mathematics is self-revealing.
I do think evolution has made us find mathematics beautiful for the same reason we find other things beautiful: geometers are survival assets and a basis for technology.
FireTag says: “I do think evolution has made us find mathematics beautiful for the same reason we find other things beautiful: geometers are survival assets and a basis for technology.”
This statement has an unavoidable logical implication — namely that ‘beauty’ is an objectively measurable thing. So why can’t we define it then?
Bruce:
The survivors DO. 😀
That’s why there are cross-cultural ratios of measurements of waist to hips in women that males find attractive. There are similar things for men, but I was not paying attention to those.
There are also mathematical and artistic connections in terms of what is beautiful. The Community of Christ Temple is built, for artistic symbolism, around a spiral that is intimately involved with the construction of a sea shell. The shell, and many other artistically “beautiful” pieces are built around the “golden ratio”, related to another of those irrational numbers, PHI, that pop up all over nature.
And my wife used to have a book of research showing how musical “beauty” had deep connections to equations that mathematicians found elegant.
I think you’ve found a very interesting post topic, indeed.
I somehow remember reading this post before…hmm…I think I commented on it then, wondering if you had read a particular Dialogue article (written by someone with a similar last name, but a different spelling). That article was also about math, but about how we know for various reasons that the vast majority of math is beyond our grasp (e.g., transcendental numbers are infinite in number to a far higher cardinality than non-transcendental numbers. Non-computable numbers are infinite to a higher cardinality to computable numbers. The fascinating thing is that somehow, even though we cannot interact with this vast majority of non-computable numbers [that’s what it means to be non-computable, I suppose], we know there is a shadowy landscape that is comprised mostly of them.)
Anyway, I’m not a mathematician, and I don’t fully remember the article, but that leads me to my comments:
That’s the funny thing…as a non-mathematician, I don’t know it works. In my heart, I may never get to the point of knowing that it’s completely reliable. Once, I wrote an article about the Monty Hall Problem — it seems that a lot of people never “get” this problem…even when it is explained to them. And yet, mathematically, it is not a mystery.
This spills over into all of the questions you asked:
1. possible answer: because we exist, and we project and perceive meaning.
2. depending on how one answers 1, it seems like that leads to an answer for 2.
3. I could say “subjective” here, but then I’d have to be very aware of the fact that well…my understanding of the Monty Hall problem or of the way to figure out pi is just as subjective, but that doesn’t negate the fact that there apparently is solid math. The subjective part is whether I am aware of it, not whether it exists.
4. Beauty is that which elicits an aesthetic response in an individual. I’m strongly leaning on the subjective here…math, your wife, and nature are all beautiful because that is *your* response when experiencing/seeing/being around them. You can quibble on whether you have the same kind of response for one as for another (in the same way that you can quibble whether the same response occurs for “love” of friend vs. romantic love, etc.,), but still there is that response.
5. No and no. At best, IF I finally figure out a mathematical thingamajig, then I might say, “Oh, that’s clever!” and be stuck with a grin on my face for the rest of the day, but it does not compute to me to link this to awe of “something” or of God.
Andrew – that dialogue article was written by my uncle. 🙂
Bruce – nice to see you here! I enjoyed the post, I just can’t clear enough room out of my head right now at this point in the semester to come up with a decent comment.
Adam, sweet!
I wouldn’t say non-computable numbers are beyond our grasp. I’d say instead that they are not useful.
Try to figure that one out.
Is being not useful mutually exclusive to being beyond our grasp?
I’ve probably overstepped my mathematical boundaries already, though
The proper answer is probably “it depends on what you mean by Math”. Choices of axioms, like choices of anything, are inventions. Theorems that follow from those choices are discoveries. Which theorems to pursue, which approaches to take within the universe of those theorems, are again inventions.
The core reality of math depends on whether you believe that all possible sets of axioms have a logical existence prior to selection. But that is not that different from asking the question of whether all possible propositions logically exist prior to anyone choosing to express one.
The broadest definition of “real” is probably “anything you can be wrong about”, and it that sense the core of mathematics, as a function of all possible axioms, is definitely real. Theorems in the abstract are foregone conclusions before anyone chooses any set of axioms to work with, and before any system that materializes a set of axioms (a computer program, for example) comes into being.
“Is being not useful mutually exclusive to being beyond our grasp? I’ve probably overstepped my mathematical boundaries already, though”
Andrew,
Actually the real question is “what do you mean by ‘grasp'”?
If what you mean is hold the full number in your head, then obviously non-computable numbers can’t be ‘grasped’ because they are infinately long. I suspect this is what you meant, so you are actually right.
But personally I’m not sure that’s the best definition of ‘grasp’ in this instance.
It seems to me that non-computable numbers are ‘graspable’ in a certain sense. For example, I could, in theory, go write a computer program to start outputing one. (The computer would technically have to allow for true random numbers, but some do.)
However, the universe would collapse into a big crunch prior to the program finishing because it’s an infinite loop.
But because the concept of non-computable nubmers isn’t that hard to ‘grasp’ I think calling them ‘ungraspable’ is probably technically correct, but a bit misleading.
I think a better description for them is ‘not useful.’ By definition, a non-computable number never has a use. If it did, such as say discovering that it happens to be the ratio of the circumference to the radius of the circle, then actually it was computable after all and can be fully comprehended via an algorithm. (See my future post on the subject for more details on this.)
FireTag,
Hey, I want to partially dedicate this series to you, my friend. Reading the Fabric of Reality gave me the idea for this series. I’m mostly pulling ideas from either that book or books I found in its bibliography. In fact, you can think of my ‘series’ as being a sort of extended books (plural) review of ideas I found interesting.
1. Why is it that made up things can really exist objectively?
Perhaps it is the converse, that things exist objectively (like circles) and it is up to us to “make up” ways to describe them. Or perhaps everything exists in an indeterminate state until we observe it.
2. Is Math (like circles or PI) “invented” or “discovered?”
The underlying principles are inherent in the world/universe around us. We only “discover” Math as ways to describe it.
3. I mentioned that this example was “beautiful” to me. Is that a subjective “in the eye of the beholder” sort of thing? Or could beauty be objectively real too?
This is “beautiful” to me, too. But I think people react differently. My wife would hate everything about this post. It caries into other areas. She appreciates music that is more regular and “emotional”. I appreciate complex music with changing time signatures, atypical chord progressions, intricate lyrics, etc. It’s almost like Math to me.
4. What do we even mean by the word “beauty” in the first place? Why is math, nature, and my wife all beautiful? Is that one word for three things or one word for one thing?
Beauty is finding “truth”. Math is discovering the fundamental. Nature is also the “fundamental”, especially compared with our lives living in an artificial world (ie. I live in a house isolated from nature, I get in my artificial car, drive to work and spend the day in a building, and repeat). And “beauty” is also a return to the fundamental. In multiple studies, a face composed of an average of faces is generally perceived as the most beautiful. Perhaps this is the “primordial” or “fundamental” face and we are all genetic mutations of that.
5. Do you feel God’s presence when you see math work like this? If not, do you at least feel awe for “something” when you see the above example?
I feel awe in the universe around me. I feel awe when I hear a beautiful piece of music. I feel awe when I walk in nature. I feel awe when I see an elegant math proof. I feel awe when I read profound statements in religious literature.
This is perhaps the fundamental core of missionary work – ascribing the “awe” someone might feel when they read the BofM to a “testimony” of the truthfulness of the LDS Church. But what does feeling the same “awe” in all of these other contexts mean?
re 12:
Bruce,
The way I was thinking of it more was that we cannot even express it in an equation/function. I may be remember the term incorrect. There is no way to talk about a non-computable number, to find the i-th term
It’s not about being infinite in length. While 1/7 is infinite in length, we can easily make an equation to talk about it (1/7). Since it is repetitive, it’s easy to find the i-th term. While pi is infinite and does *not* repeat, we can still make equations (several ways to “figure out” pi, even though they may be, as you wrote, “intractable”) to talk about it. There is always a way to find the i-th digit of pi.
But as far as I understood, non-computable numbers are not like this. So, it’s not a matter of us lacking the RESOURCES to calculate it out (e.g., intractable)…it’s that we simply can NOT do it.
When you say these numbers have no use, I don’t really dispute that. But I think the implications and the metaphor are a bit different.
Suppose any coherent description of reality must be mathematical in nature (as FireTag wrote earlier). That means our “universe” and everything within it is mathematical in nature.
But that means that this landscape of everything we know is INFINITESIMAL in the landscape of everything that exists — because non-computable numbers — that cannot be interacted with in any way (and therefore do not find use in our recognized landscape) outnumber the computable numbers by several cardinalities. It is like we are huddled in a small part of arable land, but surrounding us is a gigantic sea of inhospitable murkiness. We KNOW there is murkiness (and that there is a staggering amount), but there is absolutely no way for us to interact with it.
Again, I could really be botching up what the article was saying…
While non-computable numbers vastly outnumber computable numbers, I believe our universe takes care of this.
Simplistically, the non-computable numbers are the numbers that you can always “fit in” between any two real numbers. If the universe was infinitely granular, then you could always “go smaller” in a similar fashion.
However, because of the uncertainty principle, there is an absolute lower limit to this. Below a certain level, quantum effects predominate, adding uncertainty to the game. I believe this is the universe’s “way” of eliminating the non-computable number problem.
So, while non-computable numbers are interesting in computer theory, math theory and logic propositions, in the “real” world, the universe eliminates them.
Maybe.
I don’t think our universe “takes care” of this. Our universe (or rather, the parts we can interface with) is *computable*. We then feel inclined to say that the “real” universe has eliminated the rest, when perhaps the issue is that the “real” universe does not represent everything there is.
“But as far as I understood, non-computable numbers are not like this. So, it’s not a matter of us lacking the RESOURCES to calculate it out (e.g., intractable)…it’s that we simply can NOT do it.”
The easiest way to create a non-computable number that can’t be interacted with easily (or at all) is to start with a decimal point and then randonly select a digit after digit on for forever.
If you had such a number, but you didn’t let it go on for forever (for the sake of argument, let’s say it goes out 50 digits) then I have to call it non-computable too, because there is no algorithm that can compute it. But it’s surely not out of our reach to interact with it.
In fact, you *can* easily create a program that puts that number into memory and instantly finds the nth digit. The limit is therefore *only* resources at this point.
And we can entirely interact with it any way we wish and as much as we wish. It just isn’t that useful to do so (because it’s just a random number) so we wouldn’t bother in most cases.
Actually, the comments you are making are *spot on* to the point I’m making. You are taking the ‘other side’ of what I am saying so to speak. I think the ‘other side’ should be rigorously defended like this. So keep it up.
But let me just say this. As I understand the concepts so far, I do not agree with your assessment here: “But that means that this landscape of everything we know is INFINITESIMAL in the landscape of everything that exists”
It’s not that what you are saying is somehow strictly untrue, it’s that (to me anyhow) it doesn’t mean what you think it means.
We *do not* seem to live in a murky universe full of mostly unreachable and incomprehensible things. If that were true, science would be nearly useless and that isn’t the case.
The real truth is that we can conceive of an infinite number of non-computable numbers, but they have no value, so it doesn’t matter. Plus — I’m going to argue — an infinite number of unwieldy non-computable numbers simply don’t exist within the physical universe except via high level conceptual-only discussions like this.
So to me the real truth is that the landscape we can, in principle, know is in fact everything worth knowing and all that can physically exist anyhow.
I don’t think this follows.
Science is very good at capturing the reachable, comprehensible things. That doesn’t mean it is capturing everything.
Consider the way you refer to non-computable numbers in terms of “value” and “usefulness.” Because they are useless, because they don’t matter, because they are valueless, it doesn’t matter that science cannot do anything with them. Because science is in the business of finding things of use, matter, and value. (e.g., that which we can make predictions with, etc.,)
I don’t think I disagree with you that they simply don’t exist within the physical universe except high level conceptually. But that’s because the physical universe is defined to be mathematical functions which are computable, valuable, and useful.
I think the question becomes: are we capable at assessing value and use correctly? Are we to believe that this physical universe is *all* that is valuable or useful (just because it is valuable and useful *to us*)? Or is it that what is valuable, what is “worth knowing,” etc., are things that reflect our limitations?
“But that’s because the physical universe is defined to be mathematical functions which are computable, valuable, and useful.”
You got it!!!!!
And isn’t that a bit weird now that we think about it? Why would that be the case? How do we even know it to be true? Is it true?
“I think the question becomes: are we capable at assessing value and use correctly?”
Start with the assumption that we are NOT capable of assessing value and use correctly, then explain to me what that sentence above means.
it’s true because we literally (and physically) cannot interact with anything else.
But since we know that the computable functions are vastly outnumbered by noncomputable ones, we also know that there’s a lot more than the physical universe. And we can’t do a lot with it, and even if we could, it wouldn’t mean anything.
*That’s* what is a bit weird to me.
Funny, that doesn’t seem weird to me in the slightest. I can easily (and have easily) explained why that would be the case. That is to say, a comprehensible universe comes equiped with an explanation on why there are an infinite number of meaningless things in terms of cardinality. The reverse is not true.
I think there is a much deeper level of reality that encompasses much of this. For example: string theory suggests a ten-dimensional fabric with some dimensions “wrapped-up”, etc. Our “computable” numbers might just be what we can see/perceive, with multiple layers way beneath that.
But then it goes back to theory if it’s something we can’t measure or see. It’s like a comment on another post – perhaps the Sun is God’s hotel when He comes to earth and we just can’t see it. Perhaps it is, perhaps it isn’t. If we can’t say one way or another, it’s all theoretical.
re 21:
I guess my answer here kinda relates to my comment in 22. We simply assume that noncomputable numbers are not valuable, meaningless…and in fact, that they do not exist in the physical universe — but what if what is really the case is that there is a value or meaning to these noncomputable numbers, but we are utterly incapable of ever discerning it? I can’t tell you what that would look like, because obviously, I’m not creative enough to think of what it would look like, being limited to physical universe kinds of things.
re 23:
But that’s not all. A meaningful universe (one where science is very useful, in fact) explains that…there are far more things that science cannot do anything with (e.g., useless) in terms of cardinality.
We say it’s because those numbers are useless, but what if it’s TRUE that science, not the noncomputable numbers, is “nearly useless”, but we don’t know because we are stuck to thinking about and dealing with the physical universe that science happens to be well-fit to?
How come you guys can see comment numbers and I can’t 😦
And what the heck is ‘remove vowels’ for?
We can see comment numbers because we use standards-compliant browsers (e.g., not Internet Explorer < 8). :3
Don't press remove vowels. That is only for extreme situations to disemvowel trolls.
I actually do see your point, Andrew S. Undoubtedly if we *could* put non-computable numbers to use, we could find actual uses for them. They would probably be strange sort of uses (such as solving the halting problem), but uses none the less. Besides, I’m sure there are a lot of theorist out there that would say ‘Yeah way! Solving the halting problem would be incredibly useful!’
Also, I know a heck of a lot of religionist that place huge value on their attempts to define what God’s omniscience means. The theory of computation probably means that their definitions are self-contradictory. Yet those religionists still do ‘value’ their (possibly) meaningless and self-contradictory definition.
So in a sense I agree with you that there is a certain subjectiveness to words like ‘value.’
On the other hand, we do need to be a bit careful here. It may yet turn out to be that we *can* put non-computable numbers to use. When we call them non-computable what we really mean is that our current theory of computing can’t compute them.
But there may yet be lurking out there a new theory of computation that has yet to be discovered that will allow us to address what we currently call non-computable numbers. (They would then *become* computable numbers.)
In fact, I had planned to do a post about that because there is a notable physicist, Roger Penrose, that believes the human mind is a ‘super-turing machine’ that does in fact exceed our current theory of computation. And there is also a conceptual ‘super-turing machine’ that has been conceptually discovered (but is currently physically unimplementable and perhaps always will be unless Penrose is correct) that can exceed our current limits of computation.
So on the one hand, there is a certain sense in which you are right. It is possible that the theory of computation puts limits on what knowledge is available, just as you are suggesting. In fact, I find that an intriguing possibility to discover the limits of knowledge like that.
But I’m just not personally sure it matters much either way outside of an abstract discussion like this. That is why I feel like your characterization is a lot more like a ‘glass half empty’ point of view. Yeah, great, we are surrounded by the concept of infinite irrational numbers that have no pattern yet are infinite, so we can’t – tautologically – compute them. But I do not see that as driving to the rather bleak view you seem to be stumping for. I see it as exactly the reverse. It means that all that is real will turn out to also be all that really matters to us.
I’m IE 8
Re: being careful about lurking new theories of computation.
The author of the Dialogue article actually anticipated this (I suppose). Regardless of whether we come up with any new method of computation or any new system of mathematics, we will never be able to discover all mathematical facts, nor can we be certain that the mathematics we are doing is free of contradictions (the author had a discussion about Godel’s theorems and something else in employ to these points).
(BTW, the article was “The Limitations of Human Thought,” by Mark Nielsen in Dialogue Fall of…2008? or 2009. Whichever one volume 42 is.)
You have talked about the ramifications of a comprehensible god in the past…but what I’m saying is, “But what if, when it says things like “God’s ways are not our ways,””, it’s point out that God is *not* comprehensible?
Re: IE 8,
Go to Page, then Compatibility View Settings. Make sure that “Display all websites in compatibility view” is UNchecked. Tell me if that does anything.
“You have talked about the ramifications of a comprehensible god in the past…but what I’m saying is, “But what if, when it says things like “God’s ways are not our ways,””, it’s point out that God is *not* comprehensible?”
I’m leery of intentional explanation spoilers. If you are going to assert this, I expect you to be prepared to back it up and do a full comparison of two explanation (or one and it’s spoiler as the case might be). Since you don’t believe in God, I don’t believe you are prepared to do so.
However, the answer to your question is “there are logical ramifications either way.” God may or may not be comprehensible. But it means something either way. My question is really if people are prepared to accept those ramifications or if they are just ‘backing in’ to their point of view without thinking about it.
The authors point that we can’t discover all mathemtical facts due to Godel’s theorem is something I plan to address later because I think it has many important uses.
But people abuse Godel’s theorem to no end. It certainly does not imply that we can’t trust our math, as some people seem to think. I’m not even convinced it meaninfully means we can’t discover all mathematical truth.
What it really implies is that you can make up self referencing statement that are neither true or untrue, such as “this statement is false.” But what does that fact imply for natural language? Not much, really. That same is true for math.
Wait, so what do you want me to back up? What two things do you want me to provide explanation for? I can try to get to it after class.
I’m saying that if you want to argue “God is incomprehensible” you need to be prepared to think through the full ramifications of that statement and be prepared to accept all logical ramifications (or lack thereof) of that position and stand by them.
Simply throwing out “God might be incomprehensible” is itself a basically meaningless statement if you aren’t taking the concept seriously and trying to follow it through to it’s logical conclusions.
And it seems to me that the very first logical conclusion is that there is no point in having a logical discussion about an incomprehensible God in the first place. So if you want to have that discussion, you first need to prove to me that there is a point to having the discussion in a consistently logical way. (Which is, of course, going to turn out to be impossible.)
On the other hand, if God *is* comprehensible (at least in principle) then that implies the very things you are trying to assert. It implies possible limits to God. But those limits are (I’m arguing) not that big a deal after all, so who cares?
The fact is that I can “Godelize” God very easily. Does God know how to lift a Rock that is unliftable. Does God know how to make a perfectly round square?
Here is a direct way to Godelize God:
Can God consistently understand this sentence as true: God can’t consistently understand this sentence as true.
Note on that last one, I can understand it consistently as true but God apparently can’t. This is what Godel actually did to math, nothing more. I’m not that impressed, I’m afraid. It certainly does not suggest the bleak view you are advocating for.
By the way, Andrew, as always, fantastic comments. You are forcing me ahead of myself (nothing wrong with that.)
I probably really need to get back to work. So you may want to ‘wait and see’ what the future holds and then bring these things up again once I’ve said more.
I’m not sure we are ‘disagreeing’ at all. Rather I think I’m valuing it differently than you. You see a (to me) bleak universe made up of mostly incomprehensible things. I see a comprehensible universe that can even apparently *somewhat* comprehend (within comprehensible limits) the incomprehensible. And those things that are incomprehensible don’t see all that important. The shear number of them means nothing to me since, by definition, I’ll never bump into any of them ever.
We are discribing the same universe in very different ways, but not in mutually exclusive ways.
What I’m trying to say is something like this:
What if it’s not that God has limits, but we have the limits? (in the same way that it may not be that non-computable numbers are the ones that are limited, useless, valueless…but it is *we* who are limited in our capacity to process — and our limitation is not contingent on current flawed mathematical thinking, but is categorical?)
In this way, having logical discussion about God is — to some extent — pointless. What would be pointless (that is, the “extent”) would be trying to figure out the “noncomputable” or “transcendent” aspects of God. I think we (that is, both theists and atheists) err when we suppose that God is completely contained within the smaller portion of “computable” things. (I probably meant this rather than comprehensible/noncomprehensible).
But I think the implications are further. Saying “God is incomprehensible/noncomputable/whatever” doesn’t ever get us to “God does not exist.” in fact, statistically, it would be more likely to exist in such a way. The issue is…if God, whether in part or in whole, exists, but is so totally foreign to us…then what are we supposed to do about that?
Maybe I’m just doing a poor job of stating my position, but my position isn’t really that we live in a “bleak universe made up of mostly incomprehensible things.” After all, the incomprehensible things — I cannot interact with in any meaningful way. So, instead, every day, I interact with a profusion of meaningful, comprehensible things — because all I can perceive, by definition, are the meaningful, comprehensible things.
But the first problem is that all I can perceive is NOT all that is around.
The second problem is, given the first problem, I don’t know what I’m supposed to do about this limitation.
Bruce:
I’m sorry I missed your earlier linked post on the comprehensibility of God until now, but at least I finally understand why people were coming to my blog from a search for “Fireseed”. I’m glad you got so much from “Fabric of Reality.”
I think a more fundamental question than whether WE can comprehend God is the question you addressed in that earlier post: Can God comprehend Himself?
It’s sort of the same dilemma as posed by the old childrens’ question: “Can God create a rock so big God can’t lift it?” Is God so great even God can’t comprehend Himself?
I’m not so sure what the correct answer is, but we haven’t spent much time thinking about a God engaged in eternally EXPERIENCING Himself, not on comprehending Himself. Personally, I hope I never comprehend God except “in the limit” as in your PI example. I get bored easily.
This is precisely the issue I bumped into when I tried to think it through exactly like you just did.
I did think of one thing (when I thought this through) that you didn’t mention:
If God has comprehensible and incomprehensible (or computable and non-computable) parts, then that would imply that reason isn’t even a reliable guide in determing which parts are which.
FireTag,
I fixed your name in hopes of reducing the missed hits.
re 40,
Indeed. The guy was arguing that such truths probably would have to be garnered in spiritual means beyond linear reasoning and that our status as children of God (who is fluent in those spiritual means) gives us some sort of access to these spiritual means. The phrase “It’s only logical” will take a new meaning if/when we realize the extent of extra-rational truths (notwithstanding all of the rational ones we still work with).
It didn’t really satisfy me, but oh well.
Read my orignial post (linked above) if you haven’t already. If this is what he was saying, I’m not really saying anything different. The difference is that these ‘spiritual means’ will consistute a set of logic / computational theory also, and therefore have their own limits (though perhaps way beyond what we currently know about.)
If what one is striving for is “perfect omniscience” for God, such is probably a logical contradiction on par with lifting rocks that God can’t lift. (But of course, it’s hard to be sure since by definition I don’t comprehend it.)
But if all you want is omniscience beyond the turing limit, such may very well turn out to be possible while still being logical. It would *not* invalidate logic, it would add to it. But it would presumably still be limited in imaginable ways.
After reading what you wrote, I think there are a lot of similarities, but with the impression that the convergence of theology with physics would go only one way. Theology wouldn’t converge into physics (with physics being the “acquiring” or “prevaling” idea). Physics would have to converge into theology.
The reason would be that physics would describe maximum potential for computability prior to deification (so it could never, alone, get past a certain point). Theology would be able to discover the remaining logic.
To address certain parts of your post:
To address this, God would be comprehendable to himself, because he understands the logic of theology. He would not be comprehendable to us prior to our deification because our physics would never algorithmically compress it.
Yes, I think this would work with what the author was saying. Even more, there are a LOT more of these forces than there are of “our” forces (that is, the ones that interact with our visible matter). To this extent, I think the author would say we can DETECT spiritual forces (even if we can’t fully explain them) because we are not purely material (err…”visible” matter).
I think the difference is…the thing I’m getting from your posts is that it seems you believe that if we just get the physics part (including human reason, human logic, etc.,) developed further, then we can reach all the theological conclusions. That is where the disagreement occurs. While theology is a physics, it is not completely the physics we can determine on our own and this isn’t just a matter of time/effort.
“Physics would have to converge into theology.”
I got complaints from people because “physics” to them meant only “the laws of nature” and they couldn’t conceive it meaning anything else. In actuality, I meant “the ultimate laws of reality.” Unfortunately I never did find a better term.
Therefore what I am calling ‘physics’ is equal to what you are calling ‘the laws of theology.’
When you go on to state that ” it is not completely the physics we can determine on our own and this isn’t just a matter of time/effort” I neither agree or disagree with this. I simply do not claim to know. It is certainly conceivable that we are cut off from the ‘ultimate theory of computation’ required to comprehend God but that God is not. But given what we currently know, I have no basis to believe (or not believe) that.
However, I can make the argument that it’s better to “assume” for the sake of argument that God is fully comprehensible, even if He is not. Can you see why that would be given our discussion so far? If not, then you’ll have to wait for my future posts.
I saw that in the comments later on, btw. But I had already posted my comment before then.
My problem is that it would require *theology* be comprehensible (since physics = theology), which it really isn’t to me. So, I guess I’ll have to wait for your future posts.
I prefer to call it “ultimate physics” rather than “laws of theology” because I think those words get closer to my point. But I accept Popper’s rule that I should be prepared to use the other persons words.
Which makes you significantly different than those I tried to talk to about this in my last post. The two that argued over word use with me had no alternatives to offer (as you just did) but instead didn’t want the concept to be valid because there was no word for it that they would accept.
This is a concern I have for Popper’s rule. People can (and often do) refuse to even consider a concept by merely declaring that any word you try to use to describe it really means something else. By ‘word policing’ like this, they can effectively avoid all meaningful discussion while acting as if they are just asking for clarification.
Thank you for not doing this. It’s a pet peeve of mine. 🙂
well, Bruce, it depends on what the meaning of the word “is” is…
(BTW, were you ever able to get comment numbers to show?)
Yes, comments are now showing. Thanks for your help.
@Bruce
I just have to note that there will never be a big crunch (at least there is no evidence that there will be, while there is great evidence for an open universe ending in a “big freeze”). This was determined in 1998
http://iopscience.iop.org/1538-3881/116/3/1009/fulltext
Yeah, I know that’s the current thinking. But I figured most people don’t know that and it doesn’t actually matter anyhow to the point at the moment.
Also, the ‘big crunch’ (or lack thereof) plays into some future posts, so I was sort of being silly and foreshadowing all at the same time.
Wow, the Platonism is pretty thick in here. Not that there’s anything wrong with that. Still, there are other approaches to some of the mathematical questions that have come up in the discussion.
Constructive mathematics, for instance, does not recognize the existence of ‘non-constructible’ objects. The precise meaning of this is complicated, both philosophically and technically (http://plato.stanford.edu/entries/mathematics-constructive). ‘Constructible’ is not the same as ‘computable’, but I believe that some statements made above (e.g., “non-computable numbers vastly outnumber computable numbers”) are not true under any form of constructivism (existence of *some* non-computables is possible).
Classical logic is consistent with the Platonic view that every “proposition” (a technical term) is either true or false. We may not always know which, but the “universe” has predetermined one or the other. This is called the law of the excluded middle. Constructivism rejects it. Constructivist truth is consistent instead with the idea that a proposition is true if we can prove it is. We sometimes cannot prove a given proposition to be either true or false.
Here’s a standard example. The proposition is that it is possible to raise an irrational number to an irrational power and obtain a rational result.
Classical proof: Let s be the square root of two (which is irrational), and consider the number s^s (s to the power s):
(1) if s^s is rational, there’s our example, and we’re done.
(2a) If s^s is not rational, then it is irrational.
(2b) By the laws of exponents, (s^s)^s = s^(s*s) = s^2 = 2. Again we have an irrational number (s^s), raised to the irrational power s, with a rational result.
In classical logic, this is a correct proof. Constructively, step (2a) is not valid, and this makes philosophical constructivist sense: just because we *can’t* prove that s^s is rational does not mean we *can* prove it is irrational.
This may seem bizarre and perverse if you haven’t seen it before, and in fact it is a minority viewpoint in the philosophy of mathematics. However, constructivism didn’t arise out of sheer contrariness. It was a response to things like this:
http://en.wikipedia.org/wiki/Banach-Tarski
If that sticks in your craw, you don’t have to go all the way to constructivism to get rid of it, but you’ll have to take at least a step in that direction.
Just stopping by and this caught my eye. Just a quick comment there is a slight problem in your proof…
Clearly we know the area of a circle is going to be proportional to the square of its radius, so Area = j*r^2
You also know the perimeter of a circle is going to be proportional to the radius so Perimeter = k * r.
You have defined Pi =1/2 k for whatever k is.
What you never established was a relationship between j and k.
That is you never proved that given a circle or radius r:
Perimeter = 2*Area/r
Now you actually have everything you need because the arc of the circle is larger than the diagonal of your boxes yet smaller than the 2 sides of the box so ….
But I thought I’d point out that little gap in the argument.