Think about counting up: you start from 0. There are many decimals in between 0 and 1, actually, an infinite amount of decimals are there. So in the same way that there is no last number there is no last decimal before 1, so you would never get to it on a scale, therefore it wouldn't actually exist for any purpose. Adding on to that all math you know wouldn't exist either.Edit I didn't make this clear enough the first time if I had a number line and I was to count up using the smallest distance I could in decimals it would go on for an infinite amount. The number one wouldn't even be conceivably possible in that sense nor any other numbers. as well, this isn't a discussion on the nature of infinity or what existence is, I just used those words. understand are going off the basic definition of infinity that even a kid would get.
It is indeed true that it is impossible to count by using steps of the form:
I am currently 'at' a real number, and I will proceed to the "next" real number
for a myriad of reasons, not the least of which is the fact that there aren't any real numbers that could be described as the "next" real number.
Fortunately, when I count, I do not use such steps. The steps I use are typically
I am currently 'at' an integer, and I will proceed to the next integer
If you're paying attention to the real numbers in-between, I "pass" by them all in a single step; I don't try to pass by each one individually in separate steps.
Firstly, I apologize for a somewhat long answer to a seemingly innocent short question. Since this is primarily a philosophical question, I have removed my science hat, so here goes.
Starting with the fundamental subject of enumeration, the convention of all numbers, including integers and reals, is a result of the explicit human realization of the perceived discreteness of matter and energy forms that we witness in a wide variety of modalities of physical reality (i.e. physical nature). When we think about it, it becomes evident that there is embedded deep in physical nature a natural inclination that projects aggregative equilibrium through discretization (i.e. uniqueness preservation) of matter and energy forms. Humans and other so-called animate and inanimate life forms (yes, inanimate life forms) have evolved to exchange and utilize matter and energy in discrete packages. Therefore, it is only natural that humans have found it advantageous to explicitly enumerate the various forms of matter and energy that we encountered in physical reality. When we talk about one item of anything, say a human being, we are by definition drawing a hard boundary in space around a certain aggregate of matter what we know as a mammal, and we proceed to call the bundle of matter and energy confined within that boundary as one human being. It is an axiomatic definition that only works because it is accepted by (almost) all humans who are able to comprehend the meaning behind the definition. Again by definition, when we say one person, we are only talking about the local matter and energy within a region of 3D Euclidean space that is typically spatially localized by the human senses of vision, touch, etc. By convention, we are not talking about the radiative heat zone or electrostatic energy zone surrounding the person, nor the gravitational range of the person. Now, to complicate further the definition of numbers, if we delve inside the body of a human being, we encounter one heart, two kidneys, one brain with two hemispheres, etc, with the boundaries of all such organs being defined and agreed by human convention according to boundaries defined by human senses. When we go even deeper to the microscopic domain we begin to see a variety of individual cells and inorganic matter that collectively comprise each organ. So, where do we draw the enumeration boundaries? Where exactly does a hand end and an index finger start? It is defined only approximately by our senses and by our convention. Where exactly does an ear end and the face start? It is all a matter of mutual agreement between humans to approximately define such boundaries. Similarly, it is a matter of human agreement to define where one number ends and another number starts. If we try to be exact about everything that we enumerate, we can never come to an agreement, because one kilogram mass is really never exactly one kilogram if we want to be absolutely accurate, if such a thing was even possible. Therefore, we draw boundaries that serve our purpose, and we move on. Likewise, the arithmetic numbering scale is a convention that is agreed by consensus among humans. We should not be surprised if we encounter a new class of beings (aliens) who have devised their own weird numbering convention that is totally incomprehensible and illogical to us humans. If the numbering system makes sense to these aliens who invented it, that is all that matters from their perspective. Even our number zero is only true by convention. Can we say that there are absolutely zero tables in an empty room? How about the gravitational force exerted by a table a continent away? Surely even a table that far away ought to contribute something to our count of “tableness”, so how can the number of tables ever be exactly zero in an empty room? By the way, can a room ever be exactly and absolutely empty? Can anything be exactly and absolutely zero? It can only be so by definition and mutual agreement.
To take an example from physics, the entire field of quantum mechanics was born out of the frustration of some physicists who were unable to enumerate some particles conventionally (classically) and get a handle on the behavior of these matter and energy forms at subatomic scales because of the fuzziness of the boundaries, which led them to invent a theoretical model that describes such behavior using a probabilistic mathematical model. This was a rude wake up call for fundamental physics, but the shock waves linger on even to this day, to mystify and confuse those who are not fully prepared.
Coming back to our numbering systems, all numbering systems (integer, real, etc.) are defined by human convention. As such, numbers (and even mathematics) is a self-evident human invention that is a tool that is used by humans to manipulate and exploit the discretization property of the matter and energy entities of physical reality. It is unrealistic to claim that our numbering system is in any way an universal or absolute entity of physical reality. Boundaries of numbers are prescribed by humans. The boundary of the one person that is you, and the one person that is me, is defined by our strict human convention of what constitutes a person, but there is a part of us in our parents and a part of us in our descendants, so where do we draw exactly the boundary of the individual human unless it is solely by convention? Physical nature is immensely more simple and elegant than the cumbersome complications that we impose upon nature by our enumeration convention. We only need and utilize numbers to the extent that the practice serves our quest for survival and continuity, otherwise there would have been no need to invent numbers of any kind, including integers, reals, and the ubiquitous zero. Numbers are not something more miraculous or mysterious than the mundane anthropogenic convention that we have created them from to serve a mundane purpose.
Therefore, in direct answer to the original question, I would propose that yes, whole (integer) and real numbers, including the number zero, all actually exist as approximate topological maps of the discretization imposed by physical reality upon the human experience, but that all numbers exist solely by human convention and not as absolute entities of nature, as I explained above.
Do whole numbers other than zero actually exist?
Do positive integers other than 0 exist? Most mathematicians believe so, and for good reason, it is fair to say that an integer like 1 corresponds to a quantity in the real world, coheres with other mathematical truths, and makes number systems work.
As for the process of determining real numbers, they do not proceed the same way that positive integers do using Peano's Axioms which presumes 0 exists, and then creates successors to essentially build the natural numbers and move the definition of 'number' forward towards integers, rationals, irrationals, and therefore reals.
Reals are defined by different methods originally by men like Cantor and Dedekind, who made use of concepts of infinity and sets.
From a philosophical perspective, it is important to think of numbers not as 'things', but more like 'the results of processes'. Pi is a classical example which expresses the ratio between circumference and diameter, but can be calculated in many different ways. Pi exists as an ideal quantity conceptually, but ideal quantities possess different properties than their approximations.
Well, the question of whether numbers "exist" is kind of awkward when placed beside your intuitive understanding of what we mean when we say they "exist". This isn't a criticism; it's a perfect valid question to ask here. It just makes it tricky to disambiguate. I'll briefly explain three concepts relevant to your question, and that should clarify things.
Firstly, when most people who think mathematical objects "exist", they don't think so on the basis that we can count to them. They think so because they can define them. So arithmetic is defined, and therefore everything you can, if you believe in the existence of mathematical objects, generate from arithmetic "exists". We don't need to ever count to some large number x to say "it exists", we just know that x = x - 1 + 1 = x. that's not a formal definition btw, im just using it for illustrative purposes.
Incidentally, your intuition about counting the numbers is similar to a philosophy of maths called intuitionism, where the only truths of mathematics are those we can "construct" with a proof. Some things in mathematics, on the standard view, are true but unprovable. Intuitionism rejects these.
Finally, you've hit a really smart point about not being able to count up to 1 from zero. that's because the "reals" are uncountable. I',m sure someone else has pointed out what that means. But we don't need to define the integers from the reals, just becuase the reals are smaller. We define them separately, there's a set of reals that can't be counted, but a set of integers that can. You don't need to "build" integers from reals, and if you did, it wouldn't be a matter of counting.
Hope that helps
The issue here is not numbers, but our perception of things as whole entities.
Our perception determines there's one or two clouds, but a cloud-unit is not something that exists physically. Clouds are just optical effects caused by a certain density of water molecules in a region, which is not physically determined, but it is bounded by our mind. In other words, you could see a cloud, but somebody in a different earth position could not see it, despite looking at the same region in the sky. You can see one cloud where other observer could see five or zero.
Numbers represent ideas or concepts (to understand what a concept is and how are they acquired, see Kant's Critique of Pure Reason). One thing is the concept (zero, one, 10⁸⁴ apples, an infinite number of apples) and another is the ideal of a physical existence of apples, which is not a real fact. Apples are like clouds, except that molecules are more dense and perhaps different. Your brain tells you where an apple ends and and another begins, but that's not the physical reality. Out of our minds, everything is just a fuzzy substance-like made of quarks or strings, which are not things like in the macroscopic realm, and don't behave as things.
Numbers do exist in our brains. So, all numbers (including zero, one, -2/5, PI or infinite) can exist in your brain. I'm pretty sure that the number 928754629384 hadn't existed in your brain until I've wrote it here. Numbers do not exist physically, nor the things that they represent.
I'll leave my two cents here :)
So I had a question also, much like the OP. But mine was, "IS there such a thing as a whole number? For example, the number 1 is made up of infinite small pieces/parts/decimal places/points. So in reality everything is an infinite fraction of tiny pieces of itself, so 1, is really ALL.
There would be no reason to ever leave the number one, nor may it even be possible. If we arrived at 2 somehow, 2 of what? 2 of 1? Well, now we have another problem, you can’t have 2 things that both equal infinity; this could cause all sorts of problems.
So, instead we stand at one and theorize what two might look like, for we have decided that it is imperative we arrive at 2 as a matter of fact, based purely, on supposition.