Arithmetic and the study of arithmetic have been around for many centuries. Used by people to trade with each other, understand each others problems, build houses etc. Arithmetic is a huge part of everyday life for everyone on the planet. So why do we have arithmetical ideas and concepts? I think this is pretty simple. Arithmetic exists because we need it to live and interact with each other. A good way for us to understand each other is through arithmetic. Although it sounds like arithmetic was found by humans, there is no way that it could have been created by us.
Arithmetic is more of something that was discovered, although it already existed in the world around us. It was discovered so we can use it to figure out everyday problems and to understand the people and world around us. Later through extensive mathematics arithmetic has also become commonly used in high level mathematics where things may not relate to real life right now or sometimes never. It is crucial to understand the difference between two kinds of mathematics to really understand the question of arithmetical truths being empirically falsifiable or not.
These two contexts in which we can analyze mathematics are pure mathematics (imaginary world) and applied mathematics (the real world around us). The imaginary world is the world that is created by formulas and mathematicians to try to understand the world in a general matter with certain theories while applied mathematics deals with real world problems rather than going for a general explanation. We can make this distinction by saying that pure mathematics never really only deals with the real world when it is applied thus causing it to be used as applied mathematics.
Thus pure mathematics to a point is the cause for applied mathematics but this does not mean that pure mathematics deals with real world problems but rather might be the answer to some of the problems in the real world. I would also like to make the question about arithmetical truths might be empirically falsifiable or not clear, because there can be misunderstandings. I think the key to understand is that if an arithmetical truth is falsifiable it in no way means that the arithmetical truth is false. It just implies that there is a possibility that it might have a wrong answer or may be proven wrong in one way.
This means that it is falsifiable if it might have one wrong answer at some point in time rather than false all together. I think this is very important while dealing with these ideas because we need to understand that mathematics theorems and ideas are proven usually in an inductive sense where every result is taken to be true or else the theorem is abolished. This allows mathematical theorems and ideas to be applied to everything in the mathematical world rather than certain areas. Moving on from this idea the first context to discuss is applied mathematics or the idea that you add 2 bananas and 2 bananas and get 4 bananas.
I believe that in the real world or in applied mathematics that arithmetic may not be true always. Lets take the example above where we have 4 bananas in 2 buckets with each bucket holding 2 buckets. After we make sure this is true we take a bucket and we put all of its contents into the other bucket and get a bucket that is empty and a bucket containing the contents of both buckets from the earlier stage. Now when we look into the bucket with the contents of both buckets from the earlier stage we might assume that there are 4 bananas in there.
Logically so because 2 + 2 = 4. But maybe we count the bananas and we make a mistake and claim that there is actually 3 bananas in the bucket (remember we emptied ALL of the contents of the other bucket into this one), what does this say about arithmetic? This might seem quite unrealistic to most people as we usually can count up to 4 without making a mistake but what if we were to think of a couple million bananas A different problem might be if we were deal with animals or objects that have properties that are not arithmetical in concept.
Putting 2 animals in a bucket and 2 animals in another bucket then emptying one bucket into the other (again making sure that all of the contents are transferred) maybe a different problem. In this problem we might encounter a totally different claim. We might see that through the animals creating, there may be more than 4 animals in our bucket. But if we were to use arithmetic to deduce the answer to this question we would get 2 + 2 = 4. Do we think this is true? Of course it is not because in the end we had more than 4 animals in the bucket.
What if one of the rabbits dies would we count it the same as the dead rabbits and end up with still 4 if one of the rabbits had given birth? This is another interesting question that requires a different thought process. I think it is not really fair to consider a problem like this as a fair assessment in reality, because you are basically assigning numbers to animals and then adding them. This is not how you should be solving the problem if you were dealing with animals for example. The biggest claim against this idea is that when you are putting animals in a bucket you are not necessarily representing 2 + 2 = 4.
Since these animals can create you are actually changing the problem and must analyze it in a sense like 2x plus 2x equals 4x where the x terms would represent maybe the probability of an extra offspring or maybe the idea that they would mate or something to that effect. Thus the problem is totally different than 2 + 2 = 4. Secondly to consider if we were making a human error when adding 2 and 2 then counting them. If we counted wrong would the answer to the problem be wrong. I think not, because in reality or for most of the people in the world that can count properly the problems answer is still the same.
But considering the question I will have to agree that the truth in your head and some peoples heads might be empirically falsifiable but not false. The second context is in theoretical or pure mathematics. Where there is a formula or way of understanding for everything. All of the pure mathematics arithmetical ideas have been proven and thus there is no way of falsifying any of these claims in the pure mathematics world. I think there is no way to say that 1 + 1 = 2 without being unrealistic because we would be going into a realm that is not our own. This would be a realm that we have never encountered or been in.
There is no proof that that realm exists in any way or that actually this claim does not exist in that reality. But here I do believe that we can think about a realm that this may exist. We would be creating another abstract world if we were to think of these non-real world objects. Thus it would be unrealistic to expect real world people to understand these ideas. We should also understand that there is a logic to pure mathematics. It is logical because it makes sense for mathematicians for it to work in this way and they have worked many centuries to bring it to this point.
Pure mathematics is also physical in the sense that we can apply it through applied mathematics and use it in everyday life. But arithmetical truths cannot be both logical and physical at the same time. Another example to pure mathematics can be in so far as calculus is applied to reality, it loses its character as a logical calculus and becomes a descriptive theory which may be empirical falsifiable; and in so far as it is treated as falsifiable and if it is treated in this way it cannot be considered true and used
Even though in the beginning of the paper I claimed that arithmetical truths are discovered from my Platonist point of view I think the main reason where we encounter problems is this exact area. If people believe that arithmetical truths are actually invented then it is fair for them to consider that these truths may actually be falsifiable. These ideas may come from them being invented by humans and thus having human errors in them in the future.
Also since they are created by humans different people might create different theorems and then interpret these theorems in different ways which could cause contradiction in the future. In conclusion I think that we need to think about mathematics in two different contexts. First we need to consider arithmetic in the context of us in our real world where we are prone to making sense of things but also having human error involved in many things that we do.
Second arithmetic is something that was found many centuries ago and has been used in our world for so long that there is no real wrong to it because it is true no matter the human errors we make. Also lastly thinking about our examples and ideas in the paper would it really be arithmetic to be thinking about concepts that consider more than just arithmetic like adding rabbits that might create and not consider that they might be pregnant or something. This makes all of us ignorant to the broader picture in one sense.