Abstract Mathematics and the Real Presence of Jesus Christ

Christ is ever present in all that exists, along with His Father and their Spirit. Yet, the traditional Christian belief is that He becomes truly present in the consecrated bread and wine during a proper Eucharistic Rite. See John 6:53 for one of Christ’s own rather blunt statements on the issue. Yet again, even many who claim to be Biblical Christians as well as most non-Christians, think it absurd that Catholics and Orthodox along with some other Christians believe Jesus Christ becomes truly present on the altar, His Body in the consecrated bread and His Blood in the consecrated wine.

Yet, as a believer in the Real Presence, I know it’s hard to accept that truth because of the inability of our minds to grasp any form of being but that of substance. And, yet, we regularly deal with truths which are not necessarily attached to substance in any way. Some of those truths, including some of those found in transfinite set theory , might be such that they can’t be attached to substance as we know it. Then again, the human brain seems to be able to somehow reach such truths. (Transfinite set theory deals with various infinities, some of which are far, far larger than ‘ordinary’ infinity. It’s interesting that St. Augustine of Hippo stated an intuition that there is at least one infinity far larger than the infinity of {1,2,3,…} and that intuition came from his belief that God is far beyond even statements about infinity or eternity.)

I’m claiming that the ability of the human mind to reach such concepts as infinities of various sizes and to use those concepts is analogous to our acceptance of the Real Presence of Jesus Christ upon the altar during a proper Eucharistic Rite. The human mind itself can reach far beyond mortal and finite realms to grasp concepts. God can help us to attain to more direct contact with His pure existence — however brief our times of such contact. Most mystics have deluded themselves. They aren’t reaching out beyond ordinary reality but rather reaching aspects of Creation of which ordinary reality is a part, a thin sheet floating upon an infinitely deep ocean.

In To See a World in a Grain of Sand, I imaginatively followed the universe back through the ongoing expansion to reach the so-called Big Bang. I traveled through that point, unseen as yet even in the imaginations of mathematical physicists, to get to a new realm of created being. As we travel back in time, the observations of astronomers and the experimental results of particle physicists and the theories of theoretical physicists, allow us to speculate that matter and energy and fields collapse towards a single form of being. For example, there was a time early in the expansion of our universe when the forms of energy generating weak nuclear force and electromagnetism were one type of energy. Go back further to times of higher density, when the universe was compressed into regions far smaller than a single atom in the current state of the universe, and — according to many current theories — other forms of energy as well as matter and fields seem to melt down into something as yet beyond explicit description.

I traveled back into that realm by assuming that this process of homogenization, of collapse into a single sort of physical entity, points back to a time when no more existed than the Primordial Universe, a manifestation of the truths which God chose for Creation. God shaped this universe from the strange stuff of that Primordial Universe, no more and no less than manifested truths. In a way that we perhaps could never understand, this universe is embedded in that Primordial Universe. Manifested truths of the most glorious sort are all around us — if that’s the right way to speak of it. We need to develop the right way to speak of Creation in its wholeness.

We need to see the truth in the Thomistic claim:

Things are true.

In some of my writings, I’ve extended that claim to:

Truths are thing-like.

These are not radical claims which rose from in Medieval fever-swamp but rather a return to the partial concreteness of the line of Socratic thinkers and to the more complete concreteness of the Biblical prophets and spiritual leaders and to that of Christ Himself. It’s not an attempt to build a human system that we might have a schematic of Creation but rather an attempt to accept this world as God created it and to use the facts of this world’s physics to build our metaphysics. If God wills, we can go further to gain some understanding of Creation beyond this universe. It would seem that the Good Lord has willed us to do so and has helped us to do so.

Some sensitive and creative souls, some of those who are poets or mathematicians or physicists or contemplatives, can sense the greater truths of Creation and can even explicate some of them. The human mind can respond to our environments and to the entire universe in such a way that it becomes an encapsulation of that universe, even of the world that is the universe seen in light of God’s purposes. Somehow those tangles of brain-cells form relationships with each other and with what lies outside so that the shadows of those greater truths are seen as if through a glass darkly in many cases, but seen.

The ability of a transfinite set theorist to conjecture infinities of different sizes and to understand how they relate to each other, the ability of certain mathematicians to develop abstract algebra a century before anyone could guess it could be useful in physics (especially nuclear physics), the ability of other mathematicians to think in terms of geometries impossible in this particular universe, show us that we can see shadows of greater realities. In this, they provide us with an analogy to what happened when Jesus Christ Himself turned bread and wine into His own flesh and blood, to what happens when a Christian priest calls upon the Holy Spirit to turn the bread and wine on the altar into the flesh and blood of the Lord Jesus Christ. A richer level of being manifests itself in front of us though we can see only a shadow where that richer being, or Being, presents itself to us.

We need ways to think about and speak of the processes by which human beings can deal with numbers far larger than the infinity of ordinary infinity. True it is that Cantor’s original proof that the set of real numbers is larger than the infinite set of natural numbers (the non-negative integers) is so simply stated that it’s often presented to sophisticated high school students. Still, how can such a thought register on a brain of flesh and blood, a brain small enough to fit in a human skull? From another angle: why is it that infinity can be reached by a reasoning process possible to a finite brain? Why are such concepts as infinity so accessible to forms of reasoning which can be used to extrapolate so easily from finite concepts by way of finite steps?

How was it that Cantor was able to prove that set of real numbers is larger than the infinite set of natural numbers? The symbols which we use for numbers in conjunction with the concept of mapping forced Cantor to realize that it was provably impossible to develop a map from natural numbers to real numbers. No matter what map you use, there are real numbers which aren’t mapped. Cantor’s proof is an instance of a shadow which the richer being of the Primordial Universe casts upon our universe.

Our minds can somehow sense these shadows, accessing by proxy realms which would seem by common sense to be well beyond the reach of finite reasoning. We access those shadows of manifested truths by disciplining and training the human mind so that it can move with the grain of the universe and then with the grain of Creation as a whole. We are made of the stuff that is manifested truths, including those truths we label ‘mathematical’. Those truths are deep, deep inside of us. They are deep inside of all matter, all energy, all fields, all specks of space and time. And yet we see them by ways of shadows they cast, shadows seen and then seen more clearly as human culture develops more and better conceptual and linguistic tools.

I try to practice metaphysics as an act of submission to reality as given to a finite man of this age, a reality which we know through a very small number of transcendental truths revealed to us by our Maker and by huge mountains of empirical knowledge not yet organized. To speak truthfully is not to speak of complex systems of truths built up in the way of a skyscraper, tiny truths playing the role of I-bars and rivets. The universe isn’t a skyscraper, though we can use some useful metaphors from human technology for describing the universe and the creatures which inhabit it.

Sophisticated mathematics developed as human beings saw first that Cosmos which is radically incomplete compared to the Einsteinian universe. The mathematics developed by men with a less adequate understanding of this universe was less adequate than our current mathematics. We don’t develop mathematics to understand the universe. The universe teaches us mathematics when we interact with it in the right way, when we explore it and shape our minds to encapsulate that universe. The universe will even reveal to us in various ways those shadows of richer being.

As mere human symbols can be adequate for bringing into focus some shadows of mathematical truths which seem well beyond any finite mind, so can the Eucharistic Rite be a means of bringing into focus shadows of a far greater Truth, the truth of a God who inhabits His very Creation, who even sent His Son to take on human flesh. But this is shadow is real even as it enters our body because it’s a shadow of our Maker and our Brother. It’s a shadow He casts over some who belong to Him, a shadow He wishes to cast over all who belong to Him.