JTF.ORG Forum
General Category => General Discussion => Topic started by: Confederate Kahanist on November 14, 2010, 06:13:48 PM
-
http://www.telegraph.co.uk/science/8118823/Large-cardinals-maths-shaken-by-the-unprovable.html
In the esoteric world of mathematical logic, a dramatic discovery has been made. Previously unnoticed gaps have been found at the very heart of maths. What is more, the only way to repair these holes is with monstrous, mysterious infinities.
To understand them, we must understand what makes mathematics different from other sciences. The difference is proof.
Other scientists spend their time gathering evidence from the physical world and testing hypotheses against it. Pure maths is built using pure deduction.
But proofs have to start somewhere. For all its sophistication, mathematics is not alchemy: we cannot conjure facts from thin air. Every proof must be based on some underlying assumptions, or axioms.
And there we reach a thorny question. Even today, we do not fully understand the ordinary whole numbers 1,2,3,4,5… or the age-old ways to combine them: addition and multiplication.
Over the centuries, mathematicians have arrived at basic axioms which numbers must obey. Mostly these are simple, such as "a+b=b+a for any two numbers a and b". But when the Austrian logician Kurt Gödel turned his mind to this in 1931, he revealed a hole at the heart of our conception of numbers. His "incompleteness theorems" showed that arithmetic can never have truly solid foundations. Whatever axioms are used, there will always be gaps. There will always be facts about numbers which cannot be deduced from our chosen axioms.
Gödel's theorems showed that maths meant that mathematicians could not hope to prove every true statement: there would always be "unprovable theorems", which cannot be deduced from the usual axioms. Most known examples, it's true, will not change how you add up your shopping bill. For practical purposes, the laws of arithmetic seemed good enough.
However, as revealed in his forthcoming book, Boolean Relation Theory and Concrete Incompleteness, Harvey Friedman has discovered facts about numbers which are far more unsettling. Like Gödel's unprovable statements, they fall through the gaps between axioms. The difference is that these are no longer artificial curiosities. Friedman's theorems are "concrete", meaning they contain genuinely interesting information concerning patterns among the numbers, which must always appear once certain conditions are met. Yet, Friedman has shown, the fact that such patterns always appear does not follow from the usual laws of arithmetic.
These patterns are not yet affecting physicists or engineers, but mathematicians are having to take unprovability seriously. In the past, they just had to show whether an idea is true or false. Now, results such as Friedman's raise the awkward possibility that the standard laws of mathematics may not provide an answer.
There is one easy way to make an unprovable theorem provable: adding more axioms. But which axioms do we need? The new axioms require a hard look at one of the most contentious issues in mathematics: infinity.
For more than 100 years, mathematicians have known that there are different kinds, and sizes, of infinity. This was first shown by the 19th-century genius Georg Cantor. Cantor's discovery was that it makes sense to say that one infinite collection can be bigger than another. Infinity resembles a ladder, with the lowest rung corresponding to the most familiar level of infinity, that of the ordinary whole numbers: 1,2,3… On the next rung lives the collection of all possible infinite decimal strings, a larger uncountably infinite collection, and so on, forever.
This astonishing breakthrough raised new questions. For instance, are there even higher levels which can never be reached this way? Such enigmatic entities are known as "large cardinals". The trouble is that whether or not they exist is a question beyond the principles of mathematics. It is equally consistent that large cardinals exist and that they do not.
At least, so we thought. But, like gods descending to earth to walk among mortals, we now realise their effect can be felt among the ordinary finite numbers. In particular, the existence of large cardinals is the condition needed to tame Friedman's unprovable theorems. If their existence is assumed as an additional axiom, then it can indeed be proven that his numerical patterns must always appear when they should. But without large cardinals, no such proof is possible. Mathematicians of earlier eras would have been amazed by this invasion of arithmetic by infinite giants.
Dr Richard Elwes is the author of 'Maths 1001: Absolutely Everything That Matters in Mathematics' (Quercus Publishing)
-
This is very interesting. Perhaps, somebody can explain it "for dummies."
-
I saw a BBC special on youtube a while back that talked about how trying to understand infinity drove some people completely insane or to their deaths. It's not an easy problem to solve in the least. I think it scares people to think that at the bottom of all of this, it's like complete chaos where NOTHING can be completely proven.
Physicists ran up against a similar wall when they're trying to combine quantum physics with Newtonian physics, there are contradictions there that make it almost impossible, and possibly impossible.
-
The phenomenon turns to be chaotic once the element of probability creeps in to it.
I love that maths which calculates the sum of all the muslamics on this earth as Zero. In fact I have named it as futuromatics.
;D :laugh:
-
If a quantity is ONE on the first day of a month and if it doubles itself during the every subsequent day, then any month with 31 days has the potential to create a magic.... The total sum of these quantities on the last day of the month surpasses the total global muslamic population.
Where is that Month... ;D
The formulae is >>
Total of Quantities = (Last Quantity*2) -1
BTW, Only Iranian mathematical wizards can understand it. :laugh:
-
I would think the idea of large cardinals is in line with infinity. Whether they exist though definitely can't be proven without assuming infinite is every number. I'd be curious to see what patterns appear and why large cardinal numbers are needed to prove these patterns and bring them in line with current understanding.
The way I look at this is that, again, there are so many aspects of the universe that we can never truly understand fully. Only God in his infinite wisdom can truly understand his creation.
-
Combination of infinity units of a dimension can be a miniature unit of another dimension.
-
Combination of infinity units of a dimension can be a miniature unit of another dimension.
What do you mean 'infinity units'... That is a contradiction of terms. Something infinite is unable to be measured because it has no beginning nor an end. What are infinite units? Infinity cannot be compared to anything finite because it has no measure, in time or in space.
-
Combination of infinity units of a dimension can be a miniature unit of another dimension.
What do you mean 'infinity units'... That is a contradiction of terms. Something infinite is unable to be measured because it has no beginning nor an end. What are infinite units? Infinity cannot be compared to anything finite because it has no measure, in time or in space.
Mathematicians provide some form of "measure" for infinity. Some infinite spaces are "bigger" and even infinitely bigger than other infinite spaces. That's what the notion of "cardinal" is all about.
Take the set of integers for instance. It is an infinite space. But if you compare it with the set of real numbers, it is infinitely smaller, because between two real numbers, however "close" they may be, there is an infinity of real numbers.
A big "leap" is between infinite sets that are countable and infinite sets that are not countable.
-
Combination of infinity units of a dimension can be a miniature unit of another dimension.
What do you mean 'infinity units'... That is a contradiction of terms. Something infinite is unable to be measured because it has no beginning nor an end. What are infinite units? Infinity cannot be compared to anything finite because it has no measure, in time or in space.
Mathematicians provide some form of "measure" for infinity. Some infinite spaces are "bigger" and even infinitely bigger than other infinite spaces. That's what the notion of "cardinal" is all about.
Take the set of integers for instance. It is an infinite space. But if you compare it with the set of real numbers, it is infinitely smaller, because between two real numbers, however "close" they may be, there is an infinity of real numbers.
A big "leap" is between infinite sets that are countable and infinite sets that are not countable.
This makes no sense... One infinite set is infinite, so is another infinite set. One infinite set is not bigger than another, both are unending... How can anyone say one infinite set is bigger or larger because infinite means without beginning or end. There seems to be a conflict of terminology. I am not a master mathematician, but I have a concept of infinity, and it is not able to be compared with anything in the real world. In my understanding infinite values cannot be compared, there is no such concept of little infinity or big infinity. Both are without beginning or ending.
-
The phenomenon turns to be chaotic once the element of probability creeps in to it.
Probability is very hard to understand. I've always thought it to be a mere mathematical trick. It's an abstraction, and it doesn't make sense to imagine it as something concrete. Once I take that chance, what is "mathematical expectation" to me? Does it matter what the chances of survival are once I am living through through this concrete actualization?
-
it's kind of hard to understand for me too Muman but maybe it goes like this, is there an infinite number of "numbers" between 1 and 2, like 1.0000001, etc. Is it smaller than the infinite amount of numbers between 1 and 3, or the same size because both are infinite?
-
Combination of infinity units of a dimension can be a miniature unit of another dimension.
What do you mean 'infinity units'... That is a contradiction of terms. Something infinite is unable to be measured because it has no beginning nor an end. What are infinite units? Infinity cannot be compared to anything finite because it has no measure, in time or in space.
Mathematicians provide some form of "measure" for infinity. Some infinite spaces are "bigger" and even infinitely bigger than other infinite spaces. That's what the notion of "cardinal" is all about.
Take the set of integers for instance. It is an infinite space. But if you compare it with the set of real numbers, it is infinitely smaller, because between two real numbers, however "close" they may be, there is an infinity of real numbers.
A big "leap" is between infinite sets that are countable and infinite sets that are not countable.
This makes no sense... One infinite set is infinite, so is another infinite set. One infinite set is not bigger than another, both are unending... How can anyone say one infinite set is bigger or larger because infinite means without beginning or end. There seems to be a conflict of terminology. I am not a master mathematician, but I have a concept of infinity, and it is not able to be compared with anything in the real world. In my understanding infinite values cannot be compared, there is no such concept of little infinity or big infinity. Both are without beginning or ending.
Nothing that relates to infinity is easy to understand. This is because our intuition gained from finite sets breaks down when dealing with infinite sets. But the mathematical theory of sets does make sense.
For finite sets, counting is just forming a bijection (a one-to-one correspondence) between the set being counted and an initial segment of the positive integers. There is no notion equivalent to counting for infinite sets. While counting gives a unique result when applied to a finite set, an infinite set may be placed into a one-to-one correspondence with many different ordinal numbers depending on how one chooses to "count" (order) it.
Different characterizations of size, when extended to infinite sets, will break different "rules" which held for finite sets. Which rules are broken varies from characterization to characterization. For example, Cantor's characterization, while preserving the rule that sometimes one set is larger than another, breaks the rule that deleting an element makes the set smaller. Another characterization may preserve the rule that deleting an element makes the set smaller, but break another rule.
A set having the same cardinal number as the "small" infinite set of integers is said to be Aleph-0. The smallest infinite set larger than Aleph-0 is known as Aleph-1. Cantor's theorem states that the cardinal number of any set is lower than the cardinal number of the set of all its subsets. A corollary is that there is no highest Aleph. In other words, the collection of "infinite sizes" is itself infinite.
-
The world is finite. The universe is finite in size. And there is also such a thing as the smallest distance, measured in quantum mechanics. If the world is finite and discrete, what is the point in creating useless abstractions, such as infinity and continuity, which only confuse and breed paradox? I think these abstractions are from the Devil. I once read an article by a mathematician, who wrote that it would be perfectly possible to have a self-consistent, functional, and useful mathematics without infinities.
Someone would object: but G-d is infinite. Yes, G-d is, but the universe isn't. We are not making theories about G-d. He is beyond our understanding, and I agree with this. We are not trying to describe G-d, we are trying to describe the universe. In a way, having the concept of infinity is like trying to become G-d. Scientists are known to have satanic pride.
-
The world is finite. The universe is finite in size. And there is also such a thing as the smallest distance, measured in quantum mechanics. If the world is finite and discrete, what is the point in creating useless abstractions, such as infinity and continuity, which only confuse and breed paradox? I think these abstractions are from the Devil. I once read an article by a mathematician, who wrote that it would be perfectly possible to have a self-consistent, functional, and useful mathematics without infinities.
Someone would object: but G-d is infinite. Yes, G-d is, but the universe isn't. We are not making theories about G-d. He is beyond our understanding, and I agree with this. We are not trying to describe G-d, we are trying to describe the universe. In a way, having the concept of infinity is like trying to become G-d. Scientists are known to have satanic pride.
I strongly disagree.
1) The concepts of infinity and continuity and the theories that have developed out of them in analysis and topology are anything but useless. Anyone who has a little mathematical knowledge knows this. Do you think mathematics are useless ? Apart from their extraordinary intrinsic interest from an intellectual point of view, mathematics are at the root of all the major developments of modern science and technology. No other discipline is as accurate and as consistent as mathematics.
2) When you say that the world is finite and discrete, you make a metaphysical statement whose meaning is obscure and which cannot be proved or verified.
3) What do you mean when you say that mathematical abstractions are created by the Devil and that scientists have satanical pride ? What is evil in trying to have a better understanding of the universe ? What is evil in exercising human intelligence ? Do you think G-d gave us this wonderful ability to think with the intention that we should not use it ? Do you think G-d wants us to remain in a state of illiteracy, primitive beliefs and superstition ? Do you think that is a way to honour HaShem and fulfill the destiny that He designed for us when He created us in his own image ?
-
Getting rid of "infinities" would mean getting rid of decimals and fractions. I think that's a really bad idea.
I read this somewhere but I don't know where it originated from. Someone posted it on another message board a long time ago. If you want to move 2 feet, you must first move 1 foot. If you want to move 1 foot, you must first move 1/2 foot, if you want to move 1/2 foot, you must first move 1/4 foot, If you want to move 1/4 foot, you must first move 1/8 foot. If you want to move 1/8 foot, you must first move 1/16 foot, and so on to infinity.
It's a miracle that anything in the universe can move anywhere at all!
-
I haven't heard of Friedman's results and I am not sure I would be able to understand them even though Set theory and mathematical logic are sort of a hobby of mine. However, Philosophically, I don't think it is very surprising that "infinite entities" such as large cardinals would reflect upon seemingly more comprehensible, countable structures such as the naturals.
The thing is we use a countable language (and so also countable number of axioms) to prove theorems about cardinals. We can only access and process a finite amount of data due to our time constraints and the concept of "proof" in mathematics is also finite- it is a finite sequence of logical deductions.
So, if we can use a countable language and only a finite set of actual mathematical operations to speak about cardinals it makes sense that certain properties of cardinals will be codependant with properties of the logical language which is used in "building" them.
-
The world is finite. The universe is finite in size. And there is also such a thing as the smallest distance, measured in quantum mechanics. If the world is finite and discrete, what is the point in creating useless abstractions, such as infinity and continuity, which only confuse and breed paradox? I think these abstractions are from the Devil. I once read an article by a mathematician, who wrote that it would be perfectly possible to have a self-consistent, functional, and useful mathematics without infinities.
Someone would object: but G-d is infinite. Yes, G-d is, but the universe isn't. We are not making theories about G-d. He is beyond our understanding, and I agree with this. We are not trying to describe G-d, we are trying to describe the universe. In a way, having the concept of infinity is like trying to become G-d. Scientists are known to have satanic pride.
I strongly disagree.
1) The concepts of infinity and continuity and the theories that have developed out of them in analysis and topology are anything but useless. Anyone who has a little mathematical knowledge knows this. Do you think mathematics are useless ? Apart from their extraordinary intrinsic interest from an intellectual point of view, mathematics are at the root of all the major developments of modern science and technology. No other discipline is as accurate and as consistent as mathematics.
2) When you say that the world is finite and discrete, you make a metaphysical statement whose meaning is obscure and which cannot be proved or verified.
3) What do you mean when you say that mathematical abstractions are created by the Devil and that scientists have satanical pride ? What is evil in trying to have a better understanding of the universe ? What is evil in exercising human intelligence ? Do you think G-d gave us this wonderful ability to think with the intention that we should not use it ? Do you think G-d wants us to remain in a state of illiteracy, primitive beliefs and superstition ? Do you think that is a way to honour HaShem and fulfill the destiny that He designed for us when He created us in his own image ?
1. I actually remembered who it was who said that finite mathematics is possible and legitimate. It was Roger Penrose - a great mathematician. He certainly understands mathematics, you can't deny. He himself did not endorse finite mathematics. He just said it was viable. And he said that a group of mathematicians advocated it. When I have time, I'll look up to see whether he named specific names.
(As a side point: please don't talk to me in patronizing tones, like I am an ignoramus. I was a child prodigy in mathematics. When I got older, I saw the underlying paradoxes, and recoiled from it, abandoning mathematics completely. I have a PhD, and I have published articles on philosophy of science. Oh - and I grew up in a family of scientists. Don't automatically presume that your interlocutors are stupider of less educated than you).
2. No, this is not a metaphysical statement, this is a physical statement about the material structure of the universe - to the best of our current understanding! There is the smallest distance, and there is the biggest distance. There is the speed-of-light limit. It is the presumption of the infinitely large and infinitely small that is - for now - a metaphysical assumption.
3. To desire to have a better understanding of the universe is an unqualified good in my book. But this good impulse is invariably accompanied by the bad - pride. Scientists are really given to satanic pride, perhaps more that any other group. Two recent examples: the global warming hoax and the bad nutritional advice of high-carb/low-fat which has dominated since the 50ies and caused untold deaths from "civilizational diseases." If you read the history of how these erroneous views became dominant and the ruthless suppression of dissent, it will amount to 2 words: pride and envy.
-
And a final thought for now (I'll return later). If something is riddled with paradoxes and contradictions, it's can't be good or right. Mystery is OK. Contradiction is bad.
-
The world is finite. The universe is finite in size. And there is also such a thing as the smallest distance, measured in quantum mechanics. If the world is finite and discrete, what is the point in creating useless abstractions, such as infinity and continuity, which only confuse and breed paradox? I think these abstractions are from the Devil. I once read an article by a mathematician, who wrote that it would be perfectly possible to have a self-consistent, functional, and useful mathematics without infinities.
Someone would object: but G-d is infinite. Yes, G-d is, but the universe isn't. We are not making theories about G-d. He is beyond our understanding, and I agree with this. We are not trying to describe G-d, we are trying to describe the universe. In a way, having the concept of infinity is like trying to become G-d. Scientists are known to have satanic pride.
I strongly disagree.
1) The concepts of infinity and continuity and the theories that have developed out of them in analysis and topology are anything but useless. Anyone who has a little mathematical knowledge knows this. Do you think mathematics are useless ? Apart from their extraordinary intrinsic interest from an intellectual point of view, mathematics are at the root of all the major developments of modern science and technology. No other discipline is as accurate and as consistent as mathematics.
2) When you say that the world is finite and discrete, you make a metaphysical statement whose meaning is obscure and which cannot be proved or verified.
3) What do you mean when you say that mathematical abstractions are created by the Devil and that scientists have satanical pride ? What is evil in trying to have a better understanding of the universe ? What is evil in exercising human intelligence ? Do you think G-d gave us this wonderful ability to think with the intention that we should not use it ? Do you think G-d wants us to remain in a state of illiteracy, primitive beliefs and superstition ? Do you think that is a way to honour HaShem and fulfill the destiny that He designed for us when He created us in his own image ?
1. I actually remembered who it was who said that finite mathematics is possible and legitimate. It was Roger Penrose - a great mathematician. He certainly understands mathematics, you can't deny. He himself did not endorse finite mathematics. He just said it was viable. And he said that a group of mathematicians advocated it. When I have time, I'll look up to see whether he named specific names.
(As a side point: please don't talk to me in patronizing tones, like I am an ignoramus. I was a child prodigy in mathematics. When I got older, I saw the underlying paradoxes, and recoiled from it, abandoning mathematics completely. I have a PhD, and I have published articles on philosophy of science. Oh - and I grew up in a family of scientists. Don't automatically presume that your interlocutors are stupider of less educated than you).
2. No, this is not a metaphysical statement, this is a physical statement about the material structure of the universe - to the best of our current understanding! There is the smallest distance, and there is the biggest distance. There is the speed-of-light limit. It is the presumption of the infinitely large and infinitely small that is - for now - a metaphysical assumption.
3. To desire to have a better understanding of the universe is an unqualified good in my book. But this good impulse is invariably accompanied by the bad - pride. Scientists are really given to satanic pride, perhaps more that any other group. Two recent examples: the global warming hoax and the bad nutritional advice of high-carb/low-fat which has dominated since the 50ies and caused untold deaths from "civilizational diseases." If you read the history of how these erroneous views became dominant and the ruthless suppression of dissent, it will amount to 2 words: pride and envy.
Whether you like it or not, it is a fact that the concepts of infinity and continuity are crucial to modern mathematics. The mathematical investigation of these concepts has led to very important discoveries and has allowed to develop powerful tools, whether in calculus or in geometry, that are used in every major scientific field. Roger Penrose would certainly not deny that.
I never assume that any of my interlocutor is “stupid”. I listen to him/her first. I don’t care whether you were a “child prodigy in mathematics”, I don’t care about your PHD or any other credentials that you want to show off with, all I see is someone who says that the mathematical concepts of infinity and continuity are creations of the Devil...and I'm wondering if I'm not brought back to the 16th century, talking to a representative of the Roman Inquisition !
-
Infinity exists only as a concept. This so because all quantities can be written and all distances can be measured in one way or the other.
Infinity exists only as a concept, as a complex number exists only in a virtual and a transient manner.
Operations to the limit of Infinity does boils down to finite quantities as in calculus, again because it exists only as a concept.
Infinity may not be comparable to the Hashem and this can be understood well by solving the following equation:
Infinity*Infinity*Infinity*Zero = ? ? ?
-
Here is something I saw. I don't know whether this is what Penrose mentioned. I need to find my Penrose book.
Finite Math
http://www.sscc.edu/home/jdavidso/MathAdvising/AboutFinite.html
The word discrete helps explain where Finite Math gets its name. Discrete means broken up or separated. For example, integers are discrete objects because there are non-integer numbers in between them, but real numbers are continuous numbers because there is no identifiable separation between them.
For a maddening exercise in continuity try finding the largest real (i.e., decimal) number less than one. No, it is not 0.999999 . . . (the nines repeating forever), because it can be demonstrated that 0.999999 . . . is equal to 1. Whatever this number is it is impossible to represent it in any other than the most abstract way.
Continuity is in some ways associated with infinity and infinitesimal. Since calculus is concerned with continuous numbers and continuous functions, the subject must confront the ideas of infinity and infinitesimal. Finite Math is a subject that avoids the issues of continuity encountered in calculus, so those topics are lumped into the category of "finite mathematics."
Down with calculus!
-
Down with calculus!
How else will people find the area under a curve?
-
Oh, look, there is even such a thing as finite calculus:
http://en.wikipedia.org/wiki/Finite_difference#Calculus_of_finite_differences
-
Down with calculus!
How else will people find the area under a curve?
Rubystar, go to the link I gave in my previous message. Maybe there are alternative ways of calculation that do not involve infinity. The article on finite calculus is too advanced for me. I'll have to ask a mathematician.
-
It's interesting but it probably goes over my head too. I passed calculus a long time ago but I don't remember very much at all.
Edit: Yeah I was right, I tried looking at it and pardon the expression, but it's all "Greek" to me. ;D
-
Oh, look, there is even such a thing as finite calculus:
http://en.wikipedia.org/wiki/Finite_difference#Calculus_of_finite_differences
I'm sorry, I don't want to sound rude or aggressive, but the truth is you really don't know what you're talking about... Finite differences by no means call the mathematical concepts of infinity or continuity into question. Finite differences are a numerical method used mainly as a way to find approximate solutions to boundary value problems (i.e. partial differential equations associated with boundary conditions) on a grid. Actually, they apply to differentiable functions, and differentiability implies continuity which, in turn, implies infinity !
Likewise, "discrete" does not mean "finite". The set of the integers is discrete and it is not bounded. As I explained earlier in this thread, "discrete" refers to the notion of infinite countability.
-
Oh, look, there is even such a thing as finite calculus:
http://en.wikipedia.org/wiki/Finite_difference#Calculus_of_finite_differences
I'm sorry, I don't want to sound rude or aggressive, but the truth is you really don't know what you're talking about... Finite differences by no means call the mathematical concepts of infinity or continuity into question. Finite differences are a numerical method used mainly as a way to find approximate solutions to boundary value problems (i.e. partial differential equations associated with boundary conditions) on a grid. Actually, they apply to differentiable functions, and differentiability implies continuity which, in turn, implies infinity !
Likewise, "discrete" does not mean "finite". The set of the integers is discrete and it is not bounded. As I explained earlier in this thread, "discrete" refers to the notion of infinite countability.
I couldn't have said it better myself.
-
Oh, look, there is even such a thing as finite calculus:
http://en.wikipedia.org/wiki/Finite_difference#Calculus_of_finite_differences
I'm sorry, I don't want to sound rude or aggressive, but the truth is you really don't know what you're talking about... Finite differences by no means call the mathematical concepts of infinity or continuity into question. Finite differences are a numerical method used mainly as a way to find approximate solutions to boundary value problems (i.e. partial differential equations associated with boundary conditions) on a grid. Actually, they apply to differentiable functions, and differentiability implies continuity which, in turn, implies infinity !
Likewise, "discrete" does not mean "finite". The set of the integers is discrete and it is not bounded. As I explained earlier in this thread, "discrete" refers to the notion of infinite countability.
I know very well that "discrete" is not "finite." However re-read the text I quoted above: "Continuity is in some ways associated with infinity and infinitesimal. Since calculus is concerned with continuous numbers and continuous functions, the subject must confront the ideas of infinity and infinitesimal." The two conceptes are associated through the kinds of problems they give birth to and the ways of solving them.
Now, let's take fractals that describe the mathimatics of the real world. Fractals are discrete and not integrable. Ideally, they are infinite. But in nature, fractal shapes are obviously finite. My question is whether it is possible to describe the geometry of the natural world without an appeal to infinities. To program a fractal, you don't need an infinity. You need to stop your iterations at some finite place. A lot of the problems today are solved with computer programs which don't use infinities. I am not convinced that for every method that uses infinitity we cannot find some alternative approximate calculation that gets away from infinities. After all, everything is approximate and statistical in the real world.
Saying "you don't know what you are talking about" is a sign of a feeble mind, as any kind of an appeal to authority. I may not know "what I am talking about" as concerns specific areas, but having been trained in philosophy, I understand things in general and their interconnection.
-
Saying "you don't know what you are talking about" is a sign of a feeble mind, as any kind of an appeal to authority. I may not know "what I am talking about" as concerns specific areas, but having been trained in philosophy, I understand things in general and their interconnection.
You keep boasting about your PHD in philosophy but that does not in the least guarantee that you understand mathematics. As for calling me a "feeble mind", thanks for the compliment, but no, calling a spade a spade does not make me a feeble mind.
-
I don't want to get between the fight here. But as to the point of whether we need infinity in the "real" world:
I think there are two ways to look at this: On the one hand, we do live in a finite and discrete world. WE can only do a finite amount of calculations of finite quantities and process finite amount of data, yet they are ever growing. So the only infinite we ever need is the smallest one, countable infinite which we can never actually reach.
On the other hand, we can also say that we live in an infinite world, which contain not only infinitely large but also infinitely small elements. The simplest example of it is that it takes a continuum (that is- a greater infinite than countable) of dots to form a continuous line. It is often much easier to think of objects as continuous even if the actual calculation that we do on them with computers only do a finite approximation. Also if we don't have continuum, then we have polynomials without roots etc. And also transcendental numbers like pi and e which seem to have real world implications in physics etc wouldn't exist. Although they are finite numbers, you can never really reach them, only approximate them. But if they don't exist in the real world, then how can you have a disk of radius exactly 1, which would then have an area of exactly pi, if you don't have exactly pi.
-
I think there are two ways to look at this: On the one hand, we do live in a finite and discrete world. WE can only do a finite amount of calculations of finite quantities and process finite amount of data, yet they are ever growing. So the only infinite we ever need is the smallest one, countable infinite which we can never actually reach.
Is it the "world" that is finite and discrete or is it our perception ? It seems to me that when you refer to the amount of calculations we can do, you mean our perception.
There is also an ambiguity as to how you define "discrete". If you mean "separable", then many dimensions of reality are certainly not separable. Again, it is our biased perception that makes real objects look as if they were separated. Maybe this is so because we need this representation of reality for the purpose of practical action.
Topology is a great source of thoughts when it comes to the notion of “separability”.
On the other hand, we can also say that we live in an infinite world, which contain not only infinitely large but also infinitely small elements. The simplest example of it is that it takes a continuum (that is- a greater infinite than countable) of dots to form a continuous line.
I think one must distinguish between various dimensions of reality. Ideal objects such as a straight line exhibit properties that are very different from those shared by material objects. When you turn to ideal objects such as mathematical concepts, infinity is everywhere. But material objects are essentially bounded. Mass, heat, speed, etc. don’t go to infinity in the physical world.
It is often much easier to think of objects as continuous even if the actual calculation that we do on them with computers only do a finite approximation.
The set of exact calculations, that do not imply any finite approximation, that can be done by a human mind through mathematical relations, is infinite !
-
I think there are two ways to look at this: On the one hand, we do live in a finite and discrete world. WE can only do a finite amount of calculations of finite quantities and process finite amount of data, yet they are ever growing. So the only infinite we ever need is the smallest one, countable infinite which we can never actually reach.
Is it the "world" that is finite and discrete or is it our perception ? It seems to me that when you refer to the amount of calculations we can do, you mean our perception.
There is also an ambiguity as to how you define "discrete". If you mean "separable", then many dimensions of reality are certainly not separable. Again, it is our biased perception that makes real objects look as if they were separated. Maybe this is so because we need this representation of reality for the purpose of practical action.
Topology is a great source of thoughts when it comes to the notion of “separability”.
Our perception is certainly not finite, since we can perceive infinite things as we do in math. An of course in the pure mathematical sense infinity exists. But in the physical sense it might not exist. By "discrete" I mean a discrete topological space which means it is also separable. Why are you so sure that "many dimensions of reality are certainly not separable", do you mean space-time- how can we know if it is continuous or discrete ? We certainly are unable to prove that it is continuous because any measure we take will always have a limited accuracy. But if the space-time is discrete then it may be possible that one day we'd be able to prove it.
On the other hand, we can also say that we live in an infinite world, which contain not only infinitely large but also infinitely small elements. The simplest example of it is that it takes a continuum (that is- a greater infinite than countable) of dots to form a continuous line.
I think one must distinguish between various dimensions of reality. Ideal objects such as a straight line exhibit properties that are very different from those shared by material objects. When you turn to ideal objects such as mathematical concepts, infinity is everywhere. But material objects are essentially bounded. Mass, heat, speed, etc. don’t go to infinity in the physical world.
It is often much easier to think of objects as continuous even if the actual calculation that we do on them with computers only do a finite approximation.
The set of exact calculations, that do not imply any finite approximation, that can be done by a human mind through mathematical relations, is infinite !
Well I agree with that. It just that bounded objects still appear to be continuous (or at least not separable) and so- being made out of an infinity of infinitesimal quantities.
-
What is your highest degree, yaakov mendel?
-
What is your highest degree, yaakov mendel?
Let's leave that to the PM mailbox. I don't think most JTFers care about my educational background.
-
Imagine that when decimal system was not fully operative, the pi was considered nearest to 22/7.