Why Jesus Apologetics

(Part 1 — The Nature of Mathematical Truth)

When Arithmetic Meets Eternity

Few statements appear more certain than 1 + 1 = 2. A child can grasp it, and yet philosophers have written entire volumes proving it. In 1910, Bertrand Russell and Alfred North Whitehead devoted over three hundred pages of Principia Mathematica before finally arriving at that conclusion.¹

Why so much work for so small a truth? Because even the simplest arithmetic fact hides a deep philosophical question: Why does it hold, and what makes it necessarily true in every possible world?

If we follow that question far enough, we find ourselves standing at the edge of mathematics, looking into metaphysics.  Does the certainty of “1 + 1 = 2” emerge merely from human conventions, or does it reflect something permanent and rational at the foundation of reality?

The Logic Behind the Obvious

In everyday speech, “one plus one equals two” is almost tautological.  Yet in formal logic it is a theorem that must be built from definitions.  Peano Arithmetic, developed in the nineteenth century, begins with a handful of axioms about natural numbers: there exists a first number (0); every number has a successor; and equality obeys certain rules.  From these, addition is defined recursively so that x + 0 = x and x + S(y) = S(x + y).

Within that framework, 1 is defined as S(0) and 2 as S(S(0)); hence 1 + 1 = S(1) = 2.  The statement is true by necessity once we accept the axioms.  It cannot be false without changing what we mean by “1,” “+,” or “2.”

Mathematicians call such truths analytic—they follow from the meanings of the symbols.  They are not empirical discoveries like the boiling point of water; no experiment could reveal a universe where ordinary arithmetic behaves differently.  Even if no physical things existed to count, the truth that the successor of 1 is 2 would remain intact.

Where Do Necessary Truths Reside?

That observation raises a larger question: What kind of thing is a mathematical truth?  It seems real—after all, satellites navigate space because mathematical equations describe their orbits with uncanny accuracy.  Yet mathematical entities are not made of matter or energy.  You can’t trip over the number 2.

Philosophers have long debated three main options:

Mathematical Realism (or Platonism)

Numbers and mathematical forms exist independently of minds, in a timeless, non-physical realm.

Nominalism

Mathematics is merely a convenient linguistic construction; numbers do not literally exist.

Conceptualism (or Theistic Conceptual Realism)

Mathematical truths exist eternally, but as ideas within an eternal rational mind, namely, God’s.²

Each view tries to explain the same facts: mathematical statements are

(a) objective,

(b) necessary, and

(c) astonishingly effective in describing the physical world.

The “Unreasonable Effectiveness” of Mathematics

The physicist Eugene Wigner once called attention to the

“unreasonable effectiveness of mathematics in the natural sciences.”³

Equations dreamed up by pure mathematicians have later turned out to predict the behavior of electrons, planets, and light.  Why should an abstract language of symbols, conceived in human thought, correspond so precisely to the structure of the universe?

Naturalistic explanations often stop at pragmatism—mathematics works because we evolved to notice patterns useful for survival.  But evolution could favor rough approximations; it gives no reason the cosmos should be mathematically exact.  As John Lennox notes,

“The very fact that we can do mathematics at all—that our minds correspond to the order in nature—cries out for explanation.”⁴

This consonance between reason and reality hints that both arise from a single rational source.  It is as if the universe were written in a language already comprehensible to mind because both share the same origin in Logos—reason itself.

The Problem of Abstract Reality

If mathematical truths are timeless and independent of matter, they demand a kind of existence beyond space and time.  But abstract objects cannot act, cause, or explain anything; they simply are.  How, then, can they govern the behavior of real particles and forces?

Theologian and philosopher Alvin Plantinga argues that grounding such necessary truths in a mind avoids this problem:

“A proposition can’t exist independently of all minds, for propositions are by nature the contents of thought.”⁵ 

On this view, eternal truths are not free-floating entities but reflections of an eternal intellect.

Atheistic Platonism, by contrast, posits an impersonal realm of mathematical forms existing on their own.  Yet this realm is mysterious: it is neither mental nor physical, and it somehow structures reality without agency.  Conceptualism, the classical Christian alternative, locates these forms in God’s intellect—the divine reason that both conceives and sustains all things.

From Arithmetic to Metaphysics

We therefore arrive at a striking inference.  The necessity of 1 + 1 = 2 tells us something about the nature of reality itself.  It points to a dimension that is:

– Immaterial (not composed of matter),

– Necessary (cannot be otherwise), and

– Rational (structured by consistent logic).

If such a realm exists, and if our physical universe consistently mirrors it, then the most coherent explanation is that both originate from a rational source capable of conceiving them—what classical theism calls God.

As C.S. Lewis put it,

“Men became scientific because they expected Law in Nature, and they expected Law in Nature because they believed in a Legislator.”⁶

To Be Continued…

In Part 2, we will examine how Christian thought historically connected mathematical order with the Logos of God, explore leading atheist and naturalist objections, and show why the idea of a divine rational mind provides the most satisfying account of why “1 + 1 = 2” could never have been otherwise.

Endnotes

1. Alfred North Whitehead and Bertrand Russell, Principia Mathematica, Vol. 1 (Cambridge: Cambridge University Press, 1910), 379.

2. See William Lane Craig, “God and Abstract Objects,” in Beyond the Control of God? ed. Paul M. Gould (New York: Bloomsbury, 2014), 83–102.

3. Eugene Wigner, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” Communications in Pure and Applied Mathematics 13 (1960): 1–14.

4. John C. Lennox, Can Science Explain Everything? (Oxford: Lion Hudson, 2019), 28.

5. Alvin Plantinga, Does God Have a Nature? (Milwaukee: Marquette University Press, 1980), 3.

6. C.S. Lewis, Miracles (New York: Macmillan, 1947), 110.