It seemed obvious to me that objectivity existed. Mathematics exists, it’s objective. There is no room for opinion in mathematical problems. Could mathematics just be intersubjective belief? A language. A way to communicate physics. First, I must clarify. When I say mathematics I am not talking about the numbers the “+” sign and the fact that when I write “2+2” you think “4” I am talking about the truth statement being made, the underlying concepts where 2’s only property is its twoness, entirely abstract.
I’ll resolve the ambiguity by saying the language of mathematics does not matter. If the derivative were called the “herivative” and we used h/hx instead of d/dx, no one would know the difference; same goes for the number notation, and anything else used to communicate mathematical ideas. You can tell someone you want to buy a coffee in English or in French, you still end up with a coffee. The idea that mathematics is just a set of designated concepts that have been assigned names is called mathematical nominalism.
The problem then is only in whether mathematical concepts exist outside of one’s subjective experience. However, why stop questioning at mathematics. Does the world exist outside of our mind? Before continuing, let’s define some common metaphysical schools and possible sources of confusion.
Idealism: The world doesn’t exist outside the mind.
Solipsism: The self is all that can be know to exist.
Subjectivism: Knowledge is subjective, there is no external or objective truth.
Subjective idealism/Empirical idealism: Consciousness is the only unquestionable fact of our existence and therefore the universe is monistic. George Berkeley who is credited for the development of this theory called it immaterialism which is the metaphysical denial of the existence of a the material world.
Naive realism/Common sense realism: Our senses provide us with direct awareness of objects as they really are. Most people don’t study philosophy, they fall into this category.
Objective idealism: There is one perceiver, the perceiver is one with what they perceive. They are one with the external. Hence, accepts common sense realism.
Objectivism (Ayn Rand): “My philosophy, in essence, is the concept of man as a heroic being, with his own happiness as the moral purpose of his life, with productive achievement as his noblest activity, and reason as his only absolute.” – Ayn Rand
Objectivism (metaphysics)/Realism: That the mind acts independent from the world, that our mind makes the best approximation of such a world and the approximation can and should attract attention for improvement.
Anti-realism: The truth of a statement is determined by the logic behind it. Contrary to realism where truth depends on the independent, external world.
You may have heard the famous thought experiment: If a tree falls in the forest does it make a sound? A subjective idealist says that it does not. Think of the position this way: Everything you think of as existing outside yourself (assuming your not already an idealist) e.g. tables, cars, other people are perceived by you. To perceive is to attribute these things properties. Therefore, since a thing is perceived as having qualities, the qualities are that thing, hence it can be said to exist only inside your mind, as a collection of properties. A “quality” does not just mean a verbal description, but any kind description (i.e. audio, visual, olfactory) of every aspect that is perceived by you.
Another way to think about it is in reverse. Instead beginning with a physical tree which necessarily presupposes the existence of external things. Try to imagine what something would be like without perceiving it. What do you think of? To think of a table is to think of it’s properties. There is no escaping its properties. For me, the fact our senses may deceive us does not warrant a dismissal of the physical.
The confusion that arises when discussing the differences between physical things and how they are perceived is allayed by Kant who coined the terms “noumena” to be “a thing as it is in itself” and its antonym, “phenomena” meaning things how they are perceived. I may use these terms throughout the rest of the post.
Realism improves on naive realism by asserting that our mind only makes a best approximation rather than a direct observation of physical things. This step from naive realism to realism is small but important. It’s an admission of human fallibility or better put, human inaccuracy. We simply do not have the brain power nor enough senses to get a complete and perfect experience of the world.
Berkeley believed that since we cannot know how the images get into our brain, and since an observer is presupposed at any thought of an object, the physical world must not exist. He then extrapolates to say that this is evidence of monism, that we exist in the mind of a God who ensures alignment between my experience and yours. If I put a mug on the table, when you walk into the room, you need to be able to see the mug, or at least, that is the way we have experienced reality to this point. There is no metaphysical reason why the mug has to appear in your reality. For example, in the MMORPG World of Warcraft (I heard) players can be in the same “world” or server, talk and interact with each other, however, the landscape may be different depending on whether they have completed quest x or reached level y. I will be writing a post on “are we living inside a simulation” which will explore the consequences of the above and more in detail.
Although Berkeley’s leap towards theism is too great a leap; his solipsistic thinking, I agree with. I agree with it to the extent I agree with Descartes’ “I think therefore I am.” In that it is the only certainty. So, we can safely say that a sense of subjectiveness is confirmed. Realism is impossible to prove. However, absence of evidence is not evidence of absence.
In times where we cannot know, like I said in a previous post: it’s best to choose the decisions and beliefs that have the most utility. Here, I’m not sure which is the most helpful belief. Idealism gives you solace in the fact that ultimately, if it’s happening in your head, and your head alone, you need not worry about externalities which fits nicely with a Stoic, Buddhist, or other internally focussed moral philosophies. However, objectivism realism is helpful too. Not only is it the most easily intuited of the options, which shouldn’t be a good reason to accept it in itself, but it also allows you to differentiate between memory, dreams and hallucinations and “reality”.
I’m a solipsist in that I think only my own mental experience can be confirmed, but I’m an objectivist in that I think it to be more useful (given an assumed 50/50) to believe there is a physical or at least, external world which we perceive. The accuracy of perception is not of great importance in deciding between the two. What matters is that we perceive, and our minds are not projected on to by some Berkeleyan monistic deity or otherwise.
Objectivism also does not require any sort of collectivism. In Orwell’s 1984, he has O’Brien as an idealist or as someone who expresses idealism to enhance the effectiveness of his torture. “Collective solipsism” O’Brien called it. It designates responsibility of consistent existence between two persons to external powers, rather than higher powers, as Berkeley does. This is not a serious philosophical theory. It’s a political tool in the novel, described in this way to hyperbolise social and political control to the extent of metaphysical control.
Assuming realism, can we say that mathematics exists as an objective fact? I think so. In an objective world, mathematics (not the symbols thereof) is discovered. Not invented. The likes of Stephen Hawking, before him Albert Einstein and many others search and have searched for the unifying “Theory of Everything” in the hopes of an objective explanation for the phenomena of the universe. Since the objective already exists, the mathematics behind it can therefore not be an invention but a discovered description. When a biologist discovers a new species they do not say they have invented one, but have discovered one. They may name it, but this does not translate to invention.
The more difficult question is whether or not mathematics is objective for an idealist, not a realist as was assumed above. Even in the realists world, you perceive the mathematics. So we return back to a notion of an observer. Whether you observe a physical phenomenon or this same phenomenon is projected onto you, does not affect the truth values of the mathematical rules. If we were held together by a god as individual consciousnesses and such a god is omnipotent, the laws of nature could be changed. If such laws were constantly changed it does not mean objective rules do not exist, just that they ceaselessly change. Since such a Berkeleyan god causes you and I to see a consistent reality (we assume), it is just as possible that this god causes us to see a different reality. We have arrived at a variation of Locke’s mutant possibility; except in this variation we know that we (you and I) are mutants when juxtaposed with one another. The fact we have inconsistent realities does not really matter, in the same way that if I see blue and you see red, if we share a reality and both call it “blue” nothing changes about our overall experience. Although harder to visualise, the same could be said for laws of physics or mathematics. For a another explanation, look at Wittgenstein’s private language argument.
What about pure maths? Mathematics that has no relation to anything in the universe; maths for maths’ sake. A lot of what used to be pure maths in times past is now used to describe nature in some way. Fibonacci’s sequence is a perfect model for rabbit population growth, number theory became the basis for modern day cryptography and the list goes on. A logicist has no problem reconciling this, since to him mathematics is objective and a priori. Thus, pure mathematics does not have to be observed in nature. Until now I have been referring to the external nature of mathematics in order to say it’s objective, but it doesn’t have to be external to be objective. Pure mathematics is by definition not (yet) observed in nature. How we view the fundamental rules, the first principles which govern mathematics, necessarily dictate whether we discover or invent any mathematics, including pure mathematics. An analogy would be that a programming language. A universe coded in C may have functions called set_velocity(double mass)or create_gravity_field(planet_t *planet) which can be expressed in C. However, there are a limitless number of such functions one could create using C as a language. Since the universe is not infinitely complex, it leaves the possibility for other functions and programmatic expressions to exist that are not run by the main program. C being analogous to mathematics. Pure mathematics being analogous to “functions and programmatic expressions” that are not in the universe per se.
