GUEST ESSAY: 1 + 1 = Idealism
By J. T. Schaffer
(This is a guest essay submitted to the Metaphysical Speculations Discussion Forum, reviewed and commented on by forum members. The opinions expressed in the essay are those of its author. For my own views on the subject of this essay, see my book The Idea of the World.)
There is some question as to the ultimate nature and origin of mathematics. Does it exist fully formed in some eternal Platonic realm that we can access and learn from, or is it simply a human contrivance, similar to language, that allows us to efficiently describe some aspects of our external reality, better with the objective than the subjective?
If we argue that mathematics exists eternally and fully formed, as Plato did, we are left to explain or ignore the question of how it came to be. How is it that that which is eternal and beyond change contains that which functions to describe just the opposite? In positing an eternal realm are we abandoning reasoning to find the origin of that which represents perhaps the pinnacle of reasoning? While this might be the case, it should be accepted only as the answer of last resort if our powers of reasoning fail us.
If we assume, however, that mathematics is the product of conscious reasoning, then we need to explore how that process would work. We can start with what is perhaps the simplest mathematical fact: that 1 + 1 = 2. If we consider this fact from the perspective of Leibnitz’s Principle of the Identity of Indiscernibles, there is a problem. The Principle of the Identity of Indiscernibles states that no two distinct things can be exactly the same as each other; to be different there must be a difference. If this is true then, as a mathematical object, there can be only one number “1,” since it has no associated properties that would allow copies of itself to be considered different and unique. Accordingly, it cannot be added to itself to produce two anymore that I can add the lone dollar in my pocket to itself to get the two dollars needed to buy a cheap cup of coffee.
As a practical matter we can resolve this difficulty by considering that we are not adding one to itself but adding two unspecified objects or units of something. We can also declare that the two ones are different by fiat. And finally, we can attribute some imaginary difference to them. Regardless of the choice made, the statement 1 + 1 = 2 assumes a context that is true only for applied mathematics, not abstract.
What we need is something that allows numbers that are intrinsically the same to have some external differentiating factor. Basically this is the function that space serves. The numbers remain identical but their possible location in space can differentiate them.
If we presume that the two ones are referring to two distinct objects or units of something, the question then becomes not where does mathematics come from but: Where does that which mathematics refer to come from? The problem with this is that we end up using mathematics to describe the origination of that which gives rise to mathematics. For this to be true the two must be inextricably linked. It implies that the laws of the universe, in mathematical form, cannot be taken as a given existing prior to the Big Bang that gave rise to the universe
To declare two ones different by fiat is a functional answer. It is how the rules of any game are determined. It is how civil laws are formulated. And for an inquisitive child, it is an answer often given by parents: “Because I said so.”
For mathematics, difference by fiat makes it possible to do arithmetical operations as well as algebra. A better approach, however, would be if some hypothetical difference could be imagined. While we can add and subtract in our heads it remains a rote routine. What we need is something that allows numbers that are intrinsically the same to have some external differentiating factor. Basically this is the function that space serves. The numbers remain identical but their possible location in space can differentiate them. And this remarkable capacity of space to differentiate otherwise identical objects it not limited to mathematics, physical objects such as electrons also depend upon this remarkable capacity of space for allowing unlimited, identical copies to all coexist.
Space does more than just make copies possible. It makes number lines, graphs and, geometry possible as well. In doing so it increases our understanding of the power of mathematics to describe things. As an algebraic equation x2 + y2 = z2 can be solved for any arbitrary x and y. But to fully understand what the equation represents it is necessary to graph it in space and see that it is not just the Pythagorean theorem, but the general equation for circles. Once we have z we have a radius and can now determine the circumference of the circle and its area. The number pi is now knowable. And this pi of Euclidean geometry is the same pi that appears in many algebraic expressions that have seemingly nothing to do with space.
But as we followed this line of reasoning, we find that something unexpected happened. We found that the concept of space is required by mathematics at its most abstract but simplest level. Without it we cannot add 1 + 1 to get 2. This means that we cannot have cosmological equations that describe the expansion of the universe from a spaceless singularity unless we accept that the differences in numbers are a conscious stipulation (by fiat). This takes us back to the earlier observation that mathematics and the objects that it refers to must be inextricably linked. Yet if they are both coming into being in the same process, what is the nature of that process?
It cannot be a causal process because there is no past that precedes what we are trying to establish and describe. Causality is both past-dependent and necessarily involves an infinite regress. The alternative, acausality or randomness, is equally problematical. Acausality is non-rational. This would imply that the pinnacle of rationality would then both rest upon an irrational foundation and necessarily be an illusion.
The simplest way around this is to take the very process we are engaged in at face value and use it as our guide. We are asking questions and seeking understanding, a conscious endeavor. If we consider mathematics as a conscious construct, what is to prevent us from considering consciousness as the source of everything else as well? If the mind can conceive of mathematics, why can’t it conceive of the universe and its content as well? We have already considered that numbers can be different by fiat, a conscious choice. We have already considered that by imagining space we enhance the power of mathematics. Given this, it would not be unreasonable to consider that consciousness is the true fundamental that gives rise to mathematics and the world that it partially describes. It would explain why mathematics is better at describing the objective than the subjective.
The problem with this conclusion is that science sees consciousness arising from physicality, and religion sees consciousness arising with God. But perhaps the reason we are having trouble progressing and encountering seemingly irresolvable conflicts is that we are going in the wrong direction. There is no evidence which prevents us from considering that the subjects of both science and religion arise from consciousness itself, and then seeking to better understand that process. Doing so would put all sciences and religions on the same footing. Now that would truly be a Grand Theory of Everything, a theory of conscious, intelligent self-design driven by curiosity and the quest for self-knowledge, whose starting point is simple awareness of being.
Copyright © 2019 by J. T. Schaffer. Published with permission.