Numbers, Truth, Morality and God
What is a Number? Is it an idea or a thing? This question has been debated since Greek times, and it still remains unanswered in philosophy and science. This post examines the nature of the problem, and what its likely resolution will look like. It illustrates how the problem of numbers leads to the problem of choice, which then results in the problem of morality, which then results in questions of happiness. The resolution of these problems requires the idea of universality—namely universal ideas, universal moral principles and universal ideas about happiness. Table of ContentsRussell’s Theory of NumbersIdeas and ThingsCounting = Distinguishing + OrderingImmanuel Kant—The Role of Synthetic A PrioriHow Many Goggles?The World is a MessageNot All Messages Are TrueTruth is also A PrioriWhat Should We Choose?Truth vs. VerificationPopper’s FalsificationismScientific PragmatismThe Big HypothesisAn Introduction to Indian PhilosophyIndian and Western PerspectivesThe Necessity of GodThe Problem of Universality Russell’s Theory of Numbers Bertrand Russell and Alfred North Whitehead—two famous British philosophers—attempted to answer this question in the early 20th century as part of their mammoth three volume work entitled Principia Mathematica. They proposed a simple scheme drawn from an everyday intuitive fact about collections—they claimed that numbers are properties of collections. They are not therefore individual objects, nor are they purely ideas (supposing that ideas were in our minds and could not be objective). We can see that a collection of 5 horses has the property of fiveness, although it is not alone in having that property. The same property of fiveness, would, for instance, be seen even in the set of five cats. Russell and Whitehead therefore proposed that the number five is the property of all sets with five members, and to delineate this property, we would have to form a set whose members were sets with five members. Figure-1 Russell’s Theory of Numbers To arrive at this definition of fiveness, we would have to find all the objects in the universe and collect them into groups of five, before we could define the idea of fiveness, which presented a practical problem. However, mathematicians aren’t concerned with such practicalities. They are more interested in theoretical definitions. For instance, mathematicians frequently deal with infinite sets—e.g. the set of all rational numbers— which is an infinite set too. So the fact that this process of defining a set could potentially involve infinite steps did not concern Russell and Whitehead. There is, however, another theoretical problem in this definition. The problem is that to make a set of five cats, we must first count them as 1, 2, 3, 4, and 5. Before you can have a set of five cats, therefore, you must have the definition of ‘5’ already available. But this definition of fiveness is precisely what we are trying to arrive at through this procedure of collecting objects into sets of five! You can see that to define fiveness we must have already defined the idea of five, because otherwise we could not form any set of five members, and therefore collect these sets into a larger set called fiveness. Ideas and Things This problem in the definition of number presents a classic divide between ideas and things. We generally suppose that ideas are only in our minds—i.e. they are not real. Through science, we suppose that our mind would one day be reduced to the brain, which would then be reduced to chemistry and physics, which would then prove that the mind is nothing but matter. Once this reduction has been achieved, we would then be able to show that matter results in chemistry, which then creates the mind (through evolution and such processes), which then comes to have ideas. Figure-2 The Materialist Conception of Mind and Ideas Russell and Whitehead were trying to solve this problem for only one such idea—the idea of number. And their method of reduction essentially showed that to derive the idea of fiveness from things, the things themselves have the idea of fiveness even prior. In short, you cannot derive the idea of fiveness (or any other idea) from things without using that idea to classify and identify that thing. For instance, if you were trying to define the idea of catness by collecting all the cats in the world, you must first collect all the cats. This might be practically very difficult, but let’s assume that you can do it. However, you still have the problem of how you distinguish a cat from a horse. If you cannot distinguish the two, then you would also have horses in your set of cats, and the definition would be wrong. However, to distinguish cats from horses you must have the ideas of cats and horses even before you collect them into the sets of cats and horses to arrive at their definition! In short, the problem is not unique to the idea of numbers, but for any kind of idea whatsoever—if you try to derive the idea from things. To define an idea from a set of things, you must first collect the objects to which that idea can be applied, and therefore the idea must exist even prior. This is sometimes formally called the problem of recursion: to know something you must know it beforehand. If there were only things in the world, and our minds were byproducts of the brain, which was a byproduct of chemistry, which was a byproduct of matter, we could not know anything. To know anything from the world of things, someone must know it a priori – i.e. not from the world of things. Then, that person can impart that a priori knowledge to other people, who can then see the world of things in terms of those ideas which were previously imparted to them. To begin seeing the world in terms of those ideas, you would have to accept the idea itself upon faith. Counting = Distinguishing + Ordering Counting involves two different mechanisms—(a) a mechanism for distinguishing things from one another, and (b) a method for ordering them into a sequence. Before you can count horses as 1, 2, 3, 4, 5, you must be able to distinguish the various horses from one another. If you could not distinguish, you could not know if there were 5 horses or just 4, 3, 2 or 1. After you have been able to distinguish the horses, you must order them into a sequence—i.e. calling them the first, second, third, fourth, and fifth horse. In each of the two steps—distinguishing and ordering—you need the ideas to exist beforehand. Since this problem is not unique to counting, but exists for all ideas, you must have a rich repertoire of ideas before you can use them to know the world. The diversity that you can see in the world depends on the diversity of ideas by which you can categorize it. You cannot be born a blank slate such that the slate is written upon by the encounters with the world. You must rather be born wearing some goggles using which you can interpret—i.e. distinguish, divide and organize—the world in your mind. Figure-3 Ideological Goggles Shape Perception John Locke—another famous British philosopher—claimed that our minds are blank slates at birth. Over time, as we encounter new facts, we acquire new ideas. In other words, Locke believed that all ideas are derived from things, and can therefore—in principle—be reduced to things. The problems that we previously spoke about and which arise in counting things (and in any cognition whatsoever) show that we cannot derive ideas from things, unless we already have those ideas in us beforehand. We must be wearing a pair of ideological goggles of distinct ideas and categories to even see the world correctly in terms of those ideas and categories. In other words, the fact that we see ideas in the world is because we already had them (as our goggles) before we saw the world. A mind that is a blank slate at birth will never see anything at all. Immanuel Kant—The Role of Synthetic A Priori Immanuel Kant—an 18th century German philosopher—had a sophisticated term for this claim: he called it the synthetic a priori. The term synthetic denoted that all these ideas that exist beforehand are not trivialities—they carry very important foundations of our knowledge. The term a priori denoted that these were not derived from the external world. We had to have them if we were to know anything in the world, and therefore, we could not derive them from the world a posteriori —from experience. It is well known that some people are tone deaf while others are musically savvy. A tone deaf person cannot appreciate music; to appreciate music you must have the tonality of music in you. The tone deaf and the musically savvy therefore don’t view music in the same way: one sees music while the other doesn’t. Music teachers will tell you: you must have some music in you before you can be taught more music. You cannot start as a blank slate and be taught music because if that were the case, then every person on earth could be a fantastic musician simply by going to school. Some of us are born wearing the googles of music, while others are not. In some people, the goggles of music are crystal clear, while in others they are hazy. Accordingly, they can be great or average musicians. How Many Goggles? Once you recognize that we must be wearing goggles to see the world in terms of the ideas that are ingrained in those goggles, the very next question is: How many such goggles are possible? In other words, how many ideas must we possess a priori before we can see those ideas in the world (cognize the world in terms of those ideas)? The answer to this question is an infinity because you cannot acquire any idea from the world unless you had some preliminary form of that idea already within you. This, however, does not mean that everyone must have all the ideas beforehand. It only means that someone must have had the idea before you, who then objectivizes this idea into a thing, from which other people could also gain the idea. Before objectivizing, the creator of that object must combine the ideas themselves in his or her mind to create an abstract picture of the object to be created, which is then detailed over time to create an object. Take the preliminary idea of a car. This idea is very abstract, and it can be refined to create other concrete ideas such as SUV, hatchback, sedan, truck, sports car, etc. Once a new variation of an idea has been created, others can look at that object and acquire the idea for themselves too. Figure-4 Detailed Ideas are Produced from Abstract Ideas It is therefore correct to assume that we often acquire ideas from the external world. But that acquisition is only because someone had that idea in their minds before that idea was represented in a thing. The object you see is a vehicle or carrier of that idea, but the idea would exist (or could exist) even when all these vehicles did not. The fact that we are able to create new ideas, and then convert them into things, indicates that ideas can exist even before the things embodying them do. The ideas must therefore be logically and temporally prior to the objects in which they are embodied. We know that we cannot reduce the ideas to particular things. For instance, we cannot reduce the idea of a hatchback to a particular instance of the hatchback because if we did that reduction nothing else (apart from that particular instance) could be a hatchback. We might claim that the idea of a hatchback is a property of all the hatchbacks (past, present and future) but this claim not only presents the practical difficulty of knowing everything that exists in the past, present and future, but also the conceptual problem that to even apply the idea of a...
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