What is mathematics anyway? A system that can help us understand the world? A system of absolute truths? In this post I lay the case for their true purpose.
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What is Mathematics? As a concept, you’d be hard pressed to find someone not using it one a daily basis which tells us that this is something that comes somewhat naturally to humans.It is of course popularized and taught from a young age in the modern age, but this is not of course to mean that this is the reason for its popularity, since even unschooled people can intuitively understand and perform basic calculations.
However, this does not explain to us what Mathematics actually is and this is actually becoming a sticky point lately, as weird as that sounds. For this reason, I think we need to dispel some misconceptions about the role of mathematics in human life.
Mathematics is not descriptive
The role of mathematics is not to describe reality. In fact, mathematics cannot tell us anything useful in isolation. I cannot make any prediction at all from the equation of 1+1=2. I cannot make any conclusion, moral or empirical from any possible calculation or mathematical proof.
Rather, mathematics is explanatory. It’s role is not to provide us with knowledge, but much like logic and language, to provide us with a method to communicate ideas to other humans, or more explicitly, other brains that we expect to grasp the concept.
For example, language is not descriptive either. Me saying “this is an apple” only describes reality inasmuch as the other person understands what “this”, “is” and “apple” are. This is especially pointed in language, as it is obvious that its form is such as to utilize only the sounds the human mouth makes (and more specifically, the sounds a particular group of humans is used to make most)
And much like we can swap languages around and still communicate the same ideas, so we can swap mathematical systems around with the same effect. It is precisely because mathematics are only use to explain concepts, that the form they take does not matter, as long as those communicating use the same system.
To put it more simply: Logic, mathematics and language are not universal and external concepts, somehow outside and above reality, they are simply the means that evolved primate brains use to explain ideas to one another.
Mathematics is not a science
From the above, it naturally follows that mathematics cannot help us understand how the world works. They cannot provide us with knowledge. They can only help us express this knowledge once we have it.
However, this does not mean that they are not useful in the pursuit of knowledge. In fact, they, along with logic are immensely useful. But this is not because they are a tool that has been invented or discovered by previous generations. In fact it is a misconception to think them as such (which is why it leads to nonsense such as considering logic as proof of a deity) as they are simply the result of how human brains process information. That is to say, they are useful as much as our evolved brain is useful, as they are its result. The brain itself is the actual tool.
Certainly, mathematics take more and more complex forms, on which even more can be based on. However, all of these still do not constitute knowledge, but rather expressions of different logical concepts. That is to say, they are not useful because they give us information we did not possess before, but because they can easily transmit more complex information in a smaller package. One could even think of them as the brain’s compression capability.
The same of course applies to language with its more complex words, which only make sense if one knows a vast number of definitions required to explain them. And much like language, these compressed packets of info, can make no sense to others unless they already have the capacity (ie IQ ((Well, more appropriately I guess it should be Time*IQ)) ) to “decompress”‘ them.
Mathematics is not proof
Mathematics are axiomatic. They are like they are because we say so. Because it’s useful to have them in a particular form which other will understand. But an axiom is not a proof, it’s simply the starting points we set to start explaining the proof.
To give you an example, when I say “I put one apple in a bowl, and then I put another apple in the bowl, so I have 2 apples in the bowl now”, this is not because 1+1=2 proves it. That is simply what I used to describe how many apples I put in the bowl and how many I have in the end. It is simply used to communicate what I did.
It is nonsense to assume that anything that begins with axioms can prove or describe anything. Axioms are only useful only if there is external information which we can use them on, to discover knowledge. This is simply because they can tell us what to expect with the information we have at hand, and the deviation from this expectation, alerts us to the fact that we are missing something.
It is very important to recognise that axiomatic concepts by themselves are useless as it’s impossible to draw any conclusions without applying them to empirical observations. It is this very fine point that many seem to be missing which results in huge edifices of pure logic, which however have no relation to reality. That happens because, in order to turn an axiomatic edifice into a prescription, the ideologue needs to assume a fact, a descriptive concept for reality, and sneak that in as an immutable axiom as well. However, any assumptions that are not based in empirical testing cannot under any circumstances be considered true or unchallengeable.
Objectivism is a good example of this kind of fallacious thinking, as it tried to build itself on top of axioms (“A is A“) but in order to say anything of substance, had to assume facts out of thin air (eg “man qua man”), which of course, ended being it’s Achillean heel.
So to summarize: Mathematics do no describe reality, they merely provide the concept we need to do so. Mathematics do not provide any useful information, only the way to process it. Mathematics cannot prove or disprove anything, but can only draw our attention to missing facts, and that is only if we base them on previous proven facts, not assumptions.
Of course, one might rightly say now that this is all obvious and known. Perhaps, but in order to avoid confusions in my forthcoming posts, I think it’s important to lay some groundwork, and this should also provide an opportunity for people to point out errors in this analysis.
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"Mathematics do no describe reality, they merely provide the concept we need to do so."
Nop. One of many we can use, not _the_ concept. 😉 Sorry, but on this I have to insist.
I think I dislike mathematics because of the fact that stripping something to a processing level doesn't impress me a lot. And because of the many numbers of course. 😉
I don't see what you're saying. Say you want to explain to someone how much food to bring for a barbecue for a specific number of people. How else are you going to describe it unless using mathematics?
Well, I still use the terms "some" and "many" and "enough". That is where I do neither need nor want precision. I have been bugged with numbers most of my life, be it time or counting. And meanwhile I don't get why I should not just turn away from that. I enjoy not to be bothered one mouth more or less to feed at my party or coming over at noon, instead of precisely 11:45 – I think we make too much a system out of what surrounds us and it kind of bores me to death. 😉
But back to the point, it is ONE concept of many. Not the ONLY concept. Have you ever tried to describe a feeling, a taste or a melody with that? The most important things are not mathematicalizable, I think. That is why I am not impressed. <span class="idc-smiley"><span style="background-position: -12px 0pt;"><span>:D</span></span></span>
You've still not avoided mathematic concepts though. You're simply betting that the other person has roughly the same number in mind about "some", "many" and "enough". You'd certainly be annoyed if you asked for "many" beers and the other guy brought you 3.
But this is not about being precice, it's about the fact that mathematical concepts are useful to transfer information you want about particular issues: numerical ones. That they can be used for precision is a feature, not a necessity.
I am absolutely in agreement. My first point was that mathematics are not descriptive, they are explanatory. And even more specifically, they are explanatory only in a limited area. So they can transfer information about amounts but not about feelings. This is why we also utilize logic and language. And like you cannot describe a feeling with mathematics, you also cannot describe amounts with (non-mathematical) language
The point you're making sounds rather sophistical to me. If the axioms of a system describe reality, then the conclusions derived from those axioms describe reality as well. For example in physics we can make the assumption that for a point mass F = ma = m dv/dt. From this assumption one can predict the maximum height of a thrown softball, or the distance it will travel given its initial velocity. It would be confusion to say that predicting the distance traveled of a softball, something easily measured, is not "description of reality". The softball is not exactly a "point mass", yet the explanatory power of Newton's law + calculus is still there (if you want to get technical we are predicting the path of the ball's center-of-mass, but that's not really important).
Anon, you're missing the point that the physics equation is not an axiom, it is a proven theory which does describe reality. Mathematical axioms are nothing like this as they do not in fact describe reality but rather communicate mathematical concepts.
Anon, you're missing the point that the physics equation is not an axiom, it is a empirically proven theory which does describe reality (or specifically: How reality functions as far as we know.) Mathematical axioms are nothing like this as they do not in fact describe reality but rather communicate mathematical concepts.
To quote Einstein: “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.”
Mathematics is essentially logical reasoning about abstract objects. It has nothing to do with reality other than that we humans sometimes can abstract away from reality and create abstract objects to approximately model reality and thereby reason about reality in a systematic fashion.
I have to disagree with the phrasing: "Mathematics is not proof". Mathematics is nothing but proof. The old joke on campus was that we mathematicians were contraptions for transforming cups of coffee into theorems. Indeed, mathematics and its fundamental sub-domain of logic are the only realms of thought where anything can be truly proven. Of course, that's because at the end of the day mathematical proofs are tautological in nature as they express nothing more than logical transformations on an axiomatic basis.
However, in the real world, we have to rely upon empirical observation to supply us with our axioms, and thus we leave the realm of pure thought and can no longer truly prove much of anything as our axioms are merely abstractions and the truth of our understanding of them is always subject to the next empirical observation.
The physics equation is not an axiom, but it is not an empirically proven theory either – it is an empirically derived theory with a certain level of confidence attached to it. Gravity is not proven to exist. The next time I drop a book, it might simply float around for all we know and thereby invalidate our notion of gravity.
All of that to say that in essence I agree with this post. It is very important to distinguish between empirical evidence and pure logic. Logic and axiomatic thinking are certainly useful as is empirical observation, but a blurring of the two and failure to scrutinize one's axioms leads to nonsense like Ayn Rand's Objectivism or the absolute faith some have in Neo-Liberal Economic Theories.
I like the way you put it as well, but I still claim that it's better to think of mathematical equations as compressed information to pass rather than an abstraction from reality. I think by calling them such, you give leeway to people who would consider the "free market" or perfect competition and the like as an "abstraction of reality, useful for reasoning in a systematic fashion."