Suppose there is no such thing as infinity?

infinity is a concept

In mathematics there are an infinite amount of different infinities that are used. You could just as easily say there are no infinities if you look at things differently i.e choose to define the concept in a different way.

Our physical or mental reality which we call our universe or existence is most likely digital and finite according to my understanding of modern physics. This means things are countable. If our universe is like a movie there are only so many frames and so many pixels and colours per frame/picture.

Where infinity comes in is in terms of potential.

We have only lived so many days in ouur life and seen so many things but there might be infinite possibilities of what we might do in the future if a cure for old age were found.

The world would be exactly as it is right now…because there really isn’t such a thing as “infinity”. That which we call “infinity” is merely a mental concept, not something that has any physical reality.

When things happen such as looking at the sum of 1 + 1/2 + 1/3 + 1/4 + … (the harmonic series), we say this series diverges or the sum approaches infinity, but what we really mean is that given any really large number N the harmonic series will eventually have a sum greater than N.

Even such things as tan (π/2) are not so much “infinity” as they are “undefined”.

I’m not sure anything would be any different, and I don’t think your example makes sense.

Infinity is certainly not used in its literal sense in calculus, although many people who do not grasp the fundamentals of the limit process don’t understand this. Consider the following quote of Euler:

“Often I have considered the fact that most difficulties which block the progress of students trying to learn analysis stem from this: that although they understand little of ordinary algebra, still they attempt this more subtle art. From this it follows not only that they remain on the fringes, but in addition they entertain strange ideas about the concept of the infinite, which they must try to use.”

“Infinity” is transformed into questions of concrete existence in analysis. For example, a sequence (x_n) is said to converge to x if the sequence is ultimately contained in any open ball containing x, which means that for any epsilon > 0 there is a K(epsilon) such that if n > K(epsilon), then |x_n – x| < epsilon. Notice I never said anything about infinity here! The “limit of a series” means the limit of the sequence of partial sums and is thus defined exactly the same.

I’m not sure what you mean by infinite precision in this context. Think about the limit of the series 1 + 1/2 + 1/4 + … The limit is 2. Is 2 “infinitely precise”? It’s just a number. It just so happens that given any FINITE error margin epsilon there exists a number N such that if n > N, then |1 + 1/2 + 1/4 + … + 1/(2^(n-1)) – 2|< epsilon. No positive number epsilon > 0 is “infinitely small”–not in ordinary real analysis.

Now, there are some fields of analysis where infinitesimals and infinite quantities are considered, but you must realize again that these concepts can be rigorously defined without appealing to some sort of magical, philosophical “infinity.”

In my mind, the use of the word “infinity” is one of the main causes of misunderstanding in calculus and analysis. For example, notice the undefined and meaningless words or phrases you used: “an infinite number of terms are added”; “infinite precision”.

As a side note, many undergraduate physics textbooks are notorious for extremely sloppy use of calculus (mostly involving infinity in some way or the symbols dx and dy being treated as quantities without the formalism of tensors and differential forms), and I also hold such books accountable for widespread misunderstanding of calculus. The gradual removal of elementary analytic concepts (such as working directly with the definition of a sequence’s limit) from first courses in calculus also encourages students to “entertain strange ideas about the concept of the infinite,” as Euler put it.

Finally, I really didn’t mean for this to sound as harsh as it did. Please accept my apologies! I mean no offense by anything I’ve written.

81

I think many are missing a crucial point of your question. If infinity didn’t exist, and yet we were able to obtain arbitrarily high precision, clearly nothing would change (at least ignoring at the quantum level).

But your computer example is much more precise: here there is a fixed limit to our precision, and as a result, anything that required higher precision would suffer.

Steve

Clearly it is possible that the cosmos is finite and that both space and time are granular, in which case nothing can be infinitely numerous except mathematical sets (e.g., the set of the real numbers). Thus the question becomes, must we deem such sets “real”? To my Occamite mind the answer is no, but their reality, under a wide variety of definitions of “real,” is a highly vexed question in the philosophy of mathematics.

Source(s): Realism in Mathematics by Penelope Maddy

Not having infinite is like dividing by zero. If it happened. The world would explode.

Io answer about the impact of dropping the concept of infinity we need to think about what infinity is and what is based on.

Mathematically dividing a number by zero is undefined which makes sense since no matter how many times you add zeros you will not reach the original number.

Math is the foundations of all modern engineering and technology in addition to astronomy. When we talk about some stars that are millions of light years then it will not make a difference to us if it is one million or two millions. Even though the number is not numerically large but it is out of our range in terms of our capability and resources. It is something in the blue.

Back to dividing by zero, the term ∞ is introduced based on the concept of limits. In theory tan(π/₂) is not defined however we say ∞just because we know that tan(π/₂ -ε) is extremely large when ε is extremely small. So tan(π/₂) is defined as a virtual (hypothetical) expression called ∞.

∞ is based on the concept of limit , for that reason tan(π/₂) will be ∞ if ε is positive and -∞ if ε is negative. On other words the gradient could be -∞ or ∞ depending whether the limit is approached form the lest or the right of the y-axis in ordinary Cartesian coordinate.

This phenomenon also spots the light on another result which is -∞ and ∞ concur out of our range. Observe the tangent trigonometric function curve and see how it end at -∞ and continue from ∞ as if the function is continuous. This makes the concept of ∞ looking more vague and far from reality however consider launching two projectiles from one point going on different sense (opposite directions) assuming they will move in a constant speed and altitude. Ideally they could meet up on the other side of the world that you can not see.

This concept is more dealt with in non Euclidean geometry in which you can draw two lines both parallel to a third line from one point that does not lie on the third line. So parallel lines are believed to concur in ∞.

So. Back to your proposed mathematics, according to the logic pyramid when building geometric concept you will miss many fundamentals.

Dropping the concept of ∞ which is based on the limit, will result on abolishing calculus and analytic geometry, namely differentiation and integration. Driving the world to the middle ages when the earth was considered the core of the universe. Crippling engineering progress.

I am not sure if you will be allowed to include complex number in your proposed mathematics. Depends of you allow virtual expressions or not. Since if you do you must accept ∞ and if you don’t you must also drop complex numbers, π, infinite series, numerical analysis and much more.

There is another point I wanted to introduce which is.. what is the alternative theory that you want to introduce to compensate the absence of ∞. Non Euclidean geometry wanted to introduced new concepts that are useful for certain field which contradicts with some basic concepts. To do that some concepts in non Euclidean geometry are dropped in order to have good ground to introduce new concepts. The point is if you have a good reason for invalidating the idea of ∞ then it could be a good approach considering you introducing something more useful. However denying a useful concept such as limit with have a terrible impact on our life.

Big bang theory for example sounds like science fiction story but we accept it so far because we can’t find something better to explain the behavior of the current systems in the universe. If someone tells me it is myth because it does not appeal to him I will not agree with him however if he introduce a theory with more realistic explanations universe phenomena then I will adopt the new theory by all means.

i can believe i typed so much even though i did not go through half of what i planed introduce. i appreciate readers patience.

Conclusion … leave our ∞ alone

*************

Responding

*************

it is nice to see the asker responding to the answers creating a forum like live discussion.

OK nelson, i tried to have some sense of humor but looks like it was not funny enough. when i finish high school and enroll in university my answers may become more formal 🙂

to be more specific. my answer was inquiring about the way you want to abolish infinity.

1) logical way: abolishing the concept of infinity and everything that is built on it.

2) just denying there is something called infinity and deal with other limit based concept normally without worrying about mathematicians arguing that you have double standards.

so if you are referring to the 2nd opinion then there will be trivial impact on our life even though infinity is used as a common sense in our everyday life.

for example the sum of $120 is infinity from your primary school child’s point of view. Lambo 29 costs infinity sum of money bot $120 is not. buying 51% of Lambo is feasible to many countries treasury but building up an arsenal that defeats what USA has will cost infinity amount of resources including money.

computers do not really handle infinity in fact some PC struggled to work in order after the year 2k. computer is a stupid machine that relies on you. if you say 10 is too much then that is how it will be. if you say 10²³ mole is ordinary then that is how it will.

so. cars will still run. computers will still work and food supply will not change.

thinks about Einstein’s general theory of relativity and astronomy will not concern every day’s life

my conclusion. denying the concept of infinity will have a trivial impact on our life if not none.

I am not sufficiently equipped mathematically to answer this question, but the mind reels at the thought of it. for one thing, there would be gaps in the number line and the number line would be finite.

Infinity exists just because 0 exists..and some other reasons…

Wow….Kawther, the question said “Suppose.” No one actually refuted the fact that infinity doesn’t exist or isn’t part of reality.