Infinite Regression
I'm not sure if this is a topic for the Science section or the Religion section but here goes...
I've been wrestling with Infinite Regression for years, if not decades, and would love to hear some thoughts from some fellow philosophers on the topic.
The basic question/quandary is:
If you go back through time you eventually arrive at the Big Bang or some version thereof. And then you ask, what came before that? Ultimately, you arrive at either a) nothing, or b) something. And then you have ask the same question about that 'something'. Regardless of what that something is, the possible answers are the same, what came before that? Either a) nothing or b) something.
The result, you eventually arrive at something that has to have existed infinitely, regardless of what you call it and regardless of what attributes you give it.
And here's where things get murky. There are two main problems.
The first is infinity itself. Stepping past any sort of arguments about whether the universe around us is real or imagined, and postulating that what we see is real, all things known to us have a beginning. A beam of light starts somewhere. Even Pi begins with a calculation before it becomes into existence. A representation of infinity, the infinity symbol, cannot be drawn without placing your pen on the page and forming the figure eight. Even if you side step and try to argue that a stamp of the infinity symbol is infinite, the stamp had to get created somehow. Even the Big Bang was a beginning, and any concepts that have the Big Bang going through an infinite loop of expansion and contraction throughout existence are just theories. The point? Nothing, as far as I can tell can be shown to be a 'true' representation of infinity. So, absent a representation of infinity that is in fact infinite, we have no examples of actualy infinity, only theories, models, and symbols.
The resulting question is, if we have nothing that we can show to be infinite other than the idea of infinity, how can we fully grasp the concept of what infinity is? Yet, something must be infinite or else we would not be here. A small caveat there, something must have existed infinitly into the past at least up until the point of the Big Bang or whatever may have came before the Big Bang. There is no requirement that whatever was infinite up until that point remains in existence, however logically there no reason to believe whatever is infinite would have ceased existence.
The second problem is choice. And this one is murkier than the first. Once you arrive at the conclusion that something must be infinite the question arises either a) the infinite includes us and we are part of that infinite thing, b) we are the involuntary consequence of the existence of that infinite thing, or c) our existence is the choice of that infinite thing.
Under a) you end up with the idea of an ever expanding and contracting universe with infinite Big Bangs and some version of us or other universe emerging each time either through random creation and we are a random consequence or through the same mathematics working itself out each time and creating the same existence over and over. The problem here is that you arrive at the question why would the universe be going through this process over and over and over for all existence? There is no clear answer.
Under b) things are somewhat similar as in a) and you end up with our existence being the natural consequence of the Big Bang, meaning a big ball of burning matter will expand and eventually form galaxies, stars, planets, us, etc. But you end up with a similar question, why are planets, galaxies, and the like formed from the Big Bang. Why wouldn't the infinite universe simply remain a big ball floating in nothing and no reason to ever expand and form what we know as the universe? Even this question has the problem of if you have a big ball floating in nothing, isn't that nothing in fact 'something'?
Under c) you arrive at choice. And this option is the most interesting because here you arrive at two components. Component #1, something is infinite, and Component #2, that something made a choice for other things to exist. Choice denotes consciousness. So under this model, something is infinite and something is conscious. Without assigning any other attributes to this thing such as omnipotence or postulating whether this thing is subject to the laws of the universe or independent, or postulating whether it still exists, you stil have two key pieces of the puzzle; infinite and conscious. And, even with just that, have you not arrive at the foundation of what would define god?
My question to the forum is, do you feel this train of thought is logical? If not, what would you, a classical, or a modern philosopher insert into or change in this model? What holes do you see? Do you feel any logical leaps were made that cannot be made? Conversely, are there any leaps you feel can be made next?
I welcome any and all thoughts and discussion on the topic.
I've been wrestling with Infinite Regression for years, if not decades, and would love to hear some thoughts from some fellow philosophers on the topic.
The basic question/quandary is:
If you go back through time you eventually arrive at the Big Bang or some version thereof. And then you ask, what came before that? Ultimately, you arrive at either a) nothing, or b) something. And then you have ask the same question about that 'something'. Regardless of what that something is, the possible answers are the same, what came before that? Either a) nothing or b) something.
The result, you eventually arrive at something that has to have existed infinitely, regardless of what you call it and regardless of what attributes you give it.
And here's where things get murky. There are two main problems.
The first is infinity itself. Stepping past any sort of arguments about whether the universe around us is real or imagined, and postulating that what we see is real, all things known to us have a beginning. A beam of light starts somewhere. Even Pi begins with a calculation before it becomes into existence. A representation of infinity, the infinity symbol, cannot be drawn without placing your pen on the page and forming the figure eight. Even if you side step and try to argue that a stamp of the infinity symbol is infinite, the stamp had to get created somehow. Even the Big Bang was a beginning, and any concepts that have the Big Bang going through an infinite loop of expansion and contraction throughout existence are just theories. The point? Nothing, as far as I can tell can be shown to be a 'true' representation of infinity. So, absent a representation of infinity that is in fact infinite, we have no examples of actualy infinity, only theories, models, and symbols.
The resulting question is, if we have nothing that we can show to be infinite other than the idea of infinity, how can we fully grasp the concept of what infinity is? Yet, something must be infinite or else we would not be here. A small caveat there, something must have existed infinitly into the past at least up until the point of the Big Bang or whatever may have came before the Big Bang. There is no requirement that whatever was infinite up until that point remains in existence, however logically there no reason to believe whatever is infinite would have ceased existence.
The second problem is choice. And this one is murkier than the first. Once you arrive at the conclusion that something must be infinite the question arises either a) the infinite includes us and we are part of that infinite thing, b) we are the involuntary consequence of the existence of that infinite thing, or c) our existence is the choice of that infinite thing.
Under a) you end up with the idea of an ever expanding and contracting universe with infinite Big Bangs and some version of us or other universe emerging each time either through random creation and we are a random consequence or through the same mathematics working itself out each time and creating the same existence over and over. The problem here is that you arrive at the question why would the universe be going through this process over and over and over for all existence? There is no clear answer.
Under b) things are somewhat similar as in a) and you end up with our existence being the natural consequence of the Big Bang, meaning a big ball of burning matter will expand and eventually form galaxies, stars, planets, us, etc. But you end up with a similar question, why are planets, galaxies, and the like formed from the Big Bang. Why wouldn't the infinite universe simply remain a big ball floating in nothing and no reason to ever expand and form what we know as the universe? Even this question has the problem of if you have a big ball floating in nothing, isn't that nothing in fact 'something'?
Under c) you arrive at choice. And this option is the most interesting because here you arrive at two components. Component #1, something is infinite, and Component #2, that something made a choice for other things to exist. Choice denotes consciousness. So under this model, something is infinite and something is conscious. Without assigning any other attributes to this thing such as omnipotence or postulating whether this thing is subject to the laws of the universe or independent, or postulating whether it still exists, you stil have two key pieces of the puzzle; infinite and conscious. And, even with just that, have you not arrive at the foundation of what would define god?
My question to the forum is, do you feel this train of thought is logical? If not, what would you, a classical, or a modern philosopher insert into or change in this model? What holes do you see? Do you feel any logical leaps were made that cannot be made? Conversely, are there any leaps you feel can be made next?
I welcome any and all thoughts and discussion on the topic.
Comments (42)
Why do you say this?
Pi is the ratio of a circle's circumference to its diameter. That definition defines Pi quite accurately from any other number.
Does the number three start with calculation?
That's a pretty good question. Does it?
Wait--what happened to the (a) option--"nothing"?
Wait, this thread isn't about the regress problem? I'm disappointed (not really). Anyway, carry on. :blush:
Consider the integers as a model of time:
..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ...
Each integer has an immediate predecessor. You never arrive at "the beginning." You never ultimately arrive at either something or nothing, because you never "ultimately" arrive anywhere. You just keep on moving to the left one step at a time forever.
This example falsifies most of the bad philosophy around infinite regress.
That's otherwise known as "arriving at something" and having an infinite regress.
Pi is a number, but doesn't come into existence until calculated. It is the product of an equation. Absent the equation or at least prior to the existence of the equation it is just set of numbers without meaning. And even if it could be argued that Pi is infinite, you still have to begin it with 3. So while you could debate its existence with or without the equation, you still have to begin by writing 3, and even the equation goes through a dividing process that results in the first divide being 3. In the end it is a representation of infinity, but has a very clear beginning.
- Terrapin Station
The reference to 'nothing' meant that nothing came before the Big Bang was one of the possible choices, making the Big Bang the thing that is infinite. Whatever you arrive at, if nothing came before it, that thing is infinite. Even if we try to characterize what came before the Big Bang as 'nothing' that nothing is something, a void of white or darkness or just 'nothing' but that inself is a characteristic making it infinite, because it had to be. Essentially, everything comes from something, except that which was there at the beginning.
- fishfry
Using the integers as a example though, they cannot be shown as infinite, only represented to be infinite. Meaning to show their infinity you must first write one down on the page. Think of it as a road that goes on toward infinity to the left and right. In walking up to the road you can see the represention of its infinity but the road had to be built before it became infinite. The integers are the same as the road.
Umm... geometry doesn't need calculus. You can draw geometry, you can draw a circle, you know. With geometry you already get the mathematical constant called Pi. No computation needed.
And if with the exception of Chaitin's constant all mathematical constants are computable numbers, don't fixate on just the computability.
I think there is somewhat of a hang-up on when the numbers come into existence. Pi is not the best example to try to bring up though as even if we contend that Pi is infinite, it is only infinite in one direction. The beginning of Pi is clear and defined, meaning 3. There are no numbers to the left of Pi. For the lack of a better or more established term at the moment it can be thought of to have 'forward infinity' meaning extending into the future.
You did do a computation. You gave a finite-length description of a particular real number. The numbers for which you can do that are the computable reals.
Quoting ssu
If all you have is the computable numbers, your real line is full of holes. The computable real line is NOT a model of Euclidean geometry. For example two lines in the plane made up only of points whose coordinates are computable, may pass through each other without intersecting.
OP is making in a naive way a very subtle philosophical point that I don't believe is being adequately addressed. Defining pi using English words amounts to computing it. Hence arguments along the lines of "pi existed first, nyah nyah" all fail.
Of course pi is the output of a computation, as is the number 3. Most real numbers, and most points on Euclid's line, are not the output of any computation or finite-length description. So if people are closet Platonists, just admit that you think pi exists in God's mind before the universe was created. That's the opposite side of the proposition that pi doesn't exist before it's calculated. Let alone the noncomputables. Do they exist before there are minds to appreciate them?
Is making a drawing computation?
Quoting fishfry
Of course, but I was talking about Mathematical constants. Now there are a lot of transcendental numbers that we simply cannot define.You see a number that is transcendental and "close to" Pi isn't an accurate definition that pinpoints to one exact number.
Quoting fishfry
Exactly, but us not being able to define them (to compute them) doesn't make them not to exist.
The simply basic problem is that the mathematics we use emerges from measuring things, from arithmetic, calculation and computation. It doesn't start from it's logical roots. Hence we just have infinity as an axiom. There's a lot in Mathematics to be discovered.
How is it an approximation?
Is to you the ratio of 1/3 an approximation? As when we write it with decimal numbers in our number system its (0,33333...)
There is the problem, attributing reference to invented words that behave within a model. As is with the word, 'infinity', each assumes acceptability in its usage as reference. Words do not refer, they are behaviors, in this case behaviors acceptable to a model of world (a set of behavior with similar standards of action). These irreconcilable dualities (something - nothing, infinity - limited) show the gap between language and experience. Language (and science, a form of language) is an inventive, creative tool, not any kind of answer.
This is absolutely crazy. Irrational numbers and transcendental numbers ARE NUMBERS.
Pi is a number, it is a mathematical constant. It has an exact definition: the most general is that it's the ratio of a circle's circumference to its diameter. There are others too, actually, which hardly is suprising in mathematics.
Quoting Rank Amateur
Please read again your number theory.
I give you a board of some length. In reality that board exists, and it is an absolutely specific length.
I hand you a tape measure with 1 inch increments and ask you to measure the board. You tell me it is a little over 6 ft 1 inch long. I give you a different tape, with 1/16 inch increments. You tell me the board is 6 ft, 1 2/8 inches long, I hand you a laser tape - you tell me the board is 6ft 1.15625 inches long. Each measurement was an approximation of the the length of the board, limited by the accuracy of the tool.
Pi is a tool math uses to describe the relationship that exists in reality between the circumference and the diameter of a circle. What Pi is not is 22/7 that is an approximation, what it is not is 3.14 - what it is not is 3.141 and so on and so on. each decimal i add gets closer and makes the tool better but even if i add 1 million decimal points, we are not there, it is not as good as 1 million and one.
Pi is a good tool, a really good tool. But its use is an approximation of the reality it is trying to express.
Mathematics is a theoretical science. It's logic doesn't follow the limitations of making physical measurements.
In fact, every physical measurement is obviously an approximation. Even the number 2 is theoretical. If you want to measure exactly 2 inches, you simply cannot increase the accuracy more and more.
again, i disagree, at its base all math is, is a numerical model of reality. And it has inherent limits in its ability to do so. In many, maybe even most situations it is just a really good approximation of the reality it is trying to model. The area under the curve, is not the actual area under the curve, it is just a really really good approximation of that area and as a tool it works just fine for what we need it to do.
If you can make a drawing of a mathematical circle, more power to you. I wouldn't think I'd need to make this obvious point here. You can't define pi with a drawing. You can only define pi using a train of logical deduction in a mathematical framework. And that's a computation.
You're confusing the number pi with its decimal representation. There are plenty of numbers to the left of pi, namely {x ? ? : x < pi}. Pi does not begin with 3, its decimal representation does. Nor is pi infinite. It's a finite real number (there's no other kind) strictly between 3 and 4. Nor does a decimal expression "extend into the future." The entire decimal expression exists all at once, as a mapping from the positive integers to the set of decimal digits.
Well, the ratio of a circle's circumference to its diameter doesn't change how big or small a circle is. And I can point out what these two lengths are with a drawing. Now where this is on the number line is another question, but the ratio does stay the same.
Quoting fishfry
And Logical deduction is theoretical in math.
The problem is here that either people seem to start from decimal representation or physical measurement, which is like putting the cart in front of the horse. And this is actually a very typical way how many people think about math: that first there is reality, which is then studied with physics, hence if physics has a problem, then math has a problem. Physical measurement has it's problems when for example you go to very small scales where the problem is where one thing starts and another ends, or then in too big measurements the small size of the universe somehow comes to be the problem.
The theoretical nature of math isn't noticed.
There is a profound difference between a physical drawing, and an abstract, idealized geometric shape. You can't draw a mathematical circle with a pencil and paper. Nor could you ever make a physical measurement of any irrational number. Do you understand that? I'm asking just to make sure we're not talking past each other on this essential point.
you can't draw a mathematical circle with a pencil and paper, but do you think, if you use a compass, that the pencil end of the compass describes a mathematical circle in the process of drawing one..?
I guess maybe I would argue that there is no zero dimensional co-ordinate in the pencil end to describe the actual mathematical circle, so there is that limit, and if one argues that a planet might describe a circle like this, in its orbit, one would run up against the same problem....
No, that does not correspond to physics. A pencil point is made of chunky particles of graphite, which WIkipedia says is, "... a gray crystalline allotropic form of carbon ..." I'm not a chem major so I'll have to take their word for it. But it's certainly made of atoms, and atoms are made of quarks and gluons and whatever else atoms are made of these days. And at the bottom, according to physics, are probability waves undulating through various fields, every possible thing that can possibly happen all happening at once and then appearing to only do one thing the moment we look at it. That's modern physics.
None of this has anything to do with mathematical limits. There's no magic mathematical point at the bottom. And even if there WERE a zero-dimensional mathematical point at the end of a pencil, we could not know it! Because all physical measurement is approximate. A physical measurement is a value and a set of error bars and a probability distribution that tells you how likely it is that the value you got is actually within the error bars of the "true" value -- assuming there even is such a thing in nature.
We use math to build abstract models of the world; but the world itself is not the model. The models are just our best way of interpreting the date from the experiments we can do. And our experiments are limited by what we can build with the technology and resources we've got at any given historical moment. Science is a highly contingent enterprise.
Do not confuse the map with the territory!
yes, that was my point.
It might help some people understand a bit clearer why a pencil certainly can't either.
Yes indeed. In fact planets don't revolve in circles, which is what Copernicus believed. He gets way too much credit for getting heliocentrism right but getting the orbits wrong. It was Kepler who figured out that the orbits were ellipses; and Newton who figured out why.
But of course they are not perfect ellipses. The planets fall toward the sun. Or they are falling farther away. And they all act gravitationally on each other. So that in fact nobody knows whether the solar system is stable. Not only that, even if you knew the exact position and state of motion of each planet (which you can in classical physics) AND you perfectly modeled Newtonian gravity in a computer, you could STILL not determine the stability of the solar system. The reason is chaos. Tiny rounding errors in the computation would accumulate and be subject to the "butterfly effect." A small change in the input state leads to a large change in the output state.
https://en.wikipedia.org/wiki/Stability_of_the_Solar_System
Nothing in physical science is certain. It's just building models. The question of ultimate truth belongs to metaphysics, not physics.
maybe in Newtonian physics a planet can be modelled as orbiting in a circle if it is in a two body system.
My guess is that an equation that describes an ellipse would still be connected, or involve, pi though....so if a planet could could describe an ellipse in a mathematical way, you could derive pi from it....but that's all by the by...:)
Say what? Nothing has no properties. It's not infinite (in whatever respect you're thinking of that) or anything else.
I think you should read the Turing-Wittgenstein debates on the issue. Turing had a view similar to yours, as do I, but Wittgenstein thought it was just a convention. Neither argument is easy to defeat - it comes down to personal preference.
My belief about Pi is its defined as the ratio of a circle's circumference to its diameter. As a logical system defined by Hilbert it has a precise meaning in that system. Are logical systems eternal or only exist once discovered? Well I will let posters here discuss the issue - on one side you have Penrose - on the other Poincare and Wittgenstein.
An illumining issue to discuss in this regard is the logical principle behind Cantors diagonal argument which can be used to prove some very counter intuitive things - even Godel's Theorem. That really brings the issues with infinity to the fore.
Regarding is the universe infinite and the infinite regress argument it uses the concept of before ie of time. The modern view is the big bang was the birth of space-time. Is the concept of before valid in an era when time itself did not exist?
Thanks
Bill
Pre anything is a valid concept, as long as the "anything " exists, and that anything can certainly be space time.
Quoting Bill Hobba
The ratio of a circle to its diameter existed before it was observed, named and quantified. The ratio was not invented it was observed.
It is an absence.
Where did it exist? And what else might exist in the same realm? The baby Jesus? The Flying Spaghetti Monster? Captain Ahab?
I have trouble with Platonism.
By the way FWIW, [math]\pi[/math] is defined these days as the smallest positive zero of the sin function; which itself is defined by an infinite series or else in terms of the complex exponential. No circles are harmed or even involved.
That's your heartfelt defense of mathematical Platonism? I've seen better. Try these for a start.
https://www.iep.utm.edu/mathplat/
https://plato.stanford.edu/entries/platonism-mathematics/
Can you frame a coherent argument in support of your claim that [math]\pi[/math] has some sort (what sort?) of existence prior to the existence of any mind?
Ah. Ok! But those aren't circles. There aren't any idealized circles in nature. Mathematical abstractions are idealized versions of things in nature. Mathematical circles are inspired by circle-like things in nature, but the circle-like things aren't mathematical circles. I confess I'm surprised to have to be continually explaining this point, which I thought was universally agreed to. If you could freeze a moment in time and carefully measure the circumference and diameter of a circle of water in a pond, you would not get [math]\pi[/math] for two reasons:
1) The water's made of molecules, the molecules are made of atoms, the atoms are made of quarks and gluons and such, and the quarks and gluons themselves are just probability waves. You couldn't find a circle if you tried; and
2) All physical measurement is approximate. A physical measurement consists of some number that's the output of a physical apparatus; along with some error bars; along with a probability distribution that tells you how likely it is that the number you got is within the error bars of the "true" value, if there even is such a thing. So even if there was a perfect circle in the world (which of course there isn't) you could not measure it to arbitrary precision.
I recall writing these exact words just a day or two ago here. All physical measurement is approximate.
It's worth noting that the most accurate physical measurement we know of, something or other about the electron in quantum chromodynamics, is good to 12 decimal places. That's plenty good for a physical theory, but not very precise for a mathematical real number.
Ok. And yet ... [math]\pi[/math] is forced on us. A crude measurement of the ripple circles will give us a couple of decimal places. Archimedes calculated [math]\pi[/math] to three decimal places using mathematical reasoning.
I'm not denying the out-there-ness of [math]\pi[/math]. I'm a Platonist on weekends. But to me the case isn't as obvious as it is to others. Sure, we don't have any choice in whether 5 is prime. So it's out there. But where is it? Was it there before the Big Bang? Platonism is not as obvious as some think. How do we know that math and logic aren't qualities of our own minds and not so much of the world? Just as a bat thinks the world is full of informative sounds. We are very human-centric in these matters and I wonder if someday we'll get past that.
And you cannot ever make a physical measurement of a natural number like 2. As I've said earlier, every physical measurement is obviously an approximation. Naturally every drawing is too. Do you understand that? I guess you do, so I assume you do understand the abstract theoretical nature of mathematics, hence I'm not sure what where the disagreement is here.
Quoting fishfry
If Platonism is the problem, I think this is a different question, which is more a metaphysical one.