Success of Mathematics
In earlier epochs, people used mythological and religious narratives to encode all knowledge, even of a scientific and technological character. Words and sentences are still widely deployed in many branches.
In earlier epochs, people used mythological and religious narratives to encode all knowledge, even of a scientific and technological character. Words and sentences are still widely deployed in many branches of the Humanities, the encroachment of mathematical modeling and statistics notwithstanding. Yet, mathematics reigns supreme and unchallenged in the natural sciences. Why is that? What has catapulted mathematics (as distinct from traditional logic) to this august position within three centuries?
Mathematics is a language like no other. Still, it suffers from the drawbacks that afflict other languages. The structure of our language, its inter-relatedness with the world, and its inherent limitations dictate our worldview and determine how we understand, describe and explain Nature and our place in it. Granted, languages are living things and develop constantly (consider slang, or the emergence of infinite numbers theories in mathematics). But, they evolve within a formal grammar and syntax, a logic, a straitjacket that inhibits thinking “outside the box” and renders impossible the faithful perception of “objective” reality.
So, what made mathematics so different and so triumphant?
1. It is a universal, portable, immediately accessible language that requires no translation. Idealists would say that it is inter subjectively shared. This may be because, as Kant and others have suggested, mathematics somehow relates to or is derived from a-priori structures embedded in the human mind.
2. It provides high information density, akin to stenography. Just a few symbols arranged in formulas and equations account for a wealth of experiences and encapsulate numerous observations. Mathematical concepts and symbols do not correspond to material objects or cause them, nor do they alter reality or affect it in any way, shape, or form. One cannot map a mathematical structure or construct or number or concept into the observed universe. This is because mathematics is not confined to describing what is, or what is necessarily so – it also limns what is possible, or provable.
3. Mathematics deals with patterns and laws. It can, therefore, yield predictions. Mathematics deals with forms and structures: some of these are in the material world, others merely in the mind of the mathematician.
4. Mathematics is a flexible, “open-source”, responsive, and expandable language. Consider, for instance, how the introduction of the concept of the infinite and of infinite numbers was accommodated with relative ease despite the controversy and the threat this posed to the very foundations of traditional mathematics – or how mathematics ably progressed to deal with fuzziness and uncertainty.
5. Despite its aforementioned transience, mathematics is invariant. A mathematical advance, regardless of how arcane or revolutionary, is instantly recognizable as such and can be flawlessly incorporated in the extant body of knowledge. Thus, the fluidity of mathematics does not come at the expense of its coherence and nature.
6. There is a widespread intuition or perception that mathematics is certain because it deals with a-prior knowledge and necessary truths (either objective and “out there”, or mental, in the mind) and because it is aesthetic (like the mind of the Creator, the religious would add).
7. Finally, mathematics is useful: it works. It underlies modern science and technology unerringly and unfailingly. In time, all branches of mathematics, however obscure, prove to possess practical applications.
Still, the conundrum remains: what are mathematical objects? Do they “really” exist (the Platonic view), or are they mental figments?
Knives and forks are objects external to us. They have an objective – or at least an inter subjective – existence. Presumably, they will be there even if no one watches or uses them ever again. We can safely call them “Objective Entities”.
Our emotions and thoughts can be communicated – but they are NOT the communication itself or its contents. They are “Subjective Entities”, internal, dependent upon our existence as observers.
But what about numbers? The number one, for instance, has no objective, observer-independent status. I am not referring to the number one as adjective, as in “one apple”. I am referring to it as a stand-alone entity. As an entity it seems to stand alone in some way (it’s out there), yet be subjective in other ways (dependent upon observers). Numbers belong to a third category: “Bestowed Entities”. These are entities whose existence is bestowed upon them by social agreement between conscious agents.
But this definition is so wide that it might well be useless. Religion and money are two examples of entities which owe their existence to a social agreement between conscious entities – yet they don’t strike us as universal and out there (objective) as numbers do.
Indeed, this distinction is pertinent and our definition should be refined accordingly.
We must distinguish “Social Entities” (like money or religion) from “Bestowed Entities”. Social Entities are not universal, they are dependent on the society, culture and period that gave them birth. In contrast, numbers are Platonic ideas which come into existence through an act of conscious agreement between ALL the agents capable of reaching such an accord. While conscious agents can argue about the value of money (i.e., about its attributes) and about the existence of God – no rational, conscious agent can have an argument regarding the number one.
Apparently, the category of bestowed entities is free from the eternal dichotomy of internal versus external. It is both and comfortably so. But this is only an illusion. The dichotomy does persist. The bestowed entity is internal to the group of consenting conscious-rational agents – but it is external to any single agent (individual).
In other words, a group of rational conscious agents is certain to bestow existence on the number one. But to each and every member in the group the number one is external. It is through the power of the GROUP that existence is bestowed. From the individual’s point of view, this existence emanates from outside him (from the group) and, therefore, is external. Existence is bestowed by changing the frame of reference (from individual to group).
But this is precisely how we attribute meaning to something!!! We change our frame of reference and meaning emerges. The death of the soldier is meaningful from the point of view of the state and the rituals of the church are meaningful from the point of view of God. By shifting among frames of reference, we elicit and extract and derive meaning.
If we bestow existence and derive meaning using the same mental (cognitive) mechanism, does this mean that the two processes are one and the same? Perhaps bestowing existence is a fancy term for the more prosaic attribution of meaning? Perhaps we give meaning to a number and thereby bestow existence upon it? Perhaps the number’s existence is only its meaning and no more?
If so, all bestowed entities must be meaning-full. In other words: all of them must depend for their existence on observers (rational-conscious agents). In such a scenario, if all humans were to disappear (as well as all other intelligent observers), numbers would cease to exist.
Intuitively, we know this is not true. To prove that it is untrue is, however, difficult. Still, numbers are acknowledged to have an independent, universal quality. Their existence does depend on intelligent observers in agreement. But they exist as potentialities, as Platonic ideas, as tendencies. They materialize through the agreement of intelligent agents rather the same way that cytoplasm was supposed to have materialized through spiritualist mediums. The agreement of the group is the CHANNEL through which numbers (and other bestowed entities, such as the laws of physics) are materialized, come into being.
We are creators. In creation, one derives the new from the old. There are laws of conservation that all entities, no matter how supreme, are subject to. We can rearrange, redefine, recombine physical and other substrates. But we cannot create substrates ex nihilo. Thus, everything MUST exist one way or another before we allow it existence as we define it. This rule equally applies bestowed entities.
Wherever humans are involved, springs the eternal dichotomy of internal and external. Art makes use of a physical substrate but it succumbs to external laws of interpretation and thus derives its meaning (its existence as ART). The physical world, in contrast (similar to computer programs) contains both the substrate and the operational procedures to be applied, also known as the laws of nature.
This is the source of the conceptual confusion. In creating, we materialize that which is already there, we give it venue and allow it expression. But we are also forever bound to the dichotomy of internal and external: a HUMAN dichotomy which has to do with our false position as observers and with our ability to introspect. So, we mistakenly confuse the two issues by applying this dichotomy where it does not belong.
When we bestow existence upon a number it is not that the number is external to us and we internalize it or that it is internal and we merely externalize it. It is both external and internal. By bestowing existence upon it, we merely recognize it. In other words, it cannot be that, through interaction with us, the number changes its nature (from external to internal or the converse).
By merely realizing something and acknowledging this newfound knowledge, we do not change its nature. This is why meaning has nothing to do with existence, bestowed or not. Meaning is a human category. It is the name we give to the cognitive experience of shifting frames of reference. It has nothing to do with entities, only with us.
The world has no internal and external to it. Only we do. And when we bestow existence upon a number we only acknowledge its existence. It exists either as neural networks in our brains, or as some other entity (Platonic Idea). But, it exists and no amount of interactions with us, humans Free Reprint Articles, is ever going to change this.