Analogy Between Presuppositional Apologetics and Mathematical Proof

Christian apologetics is the discipline of giving reasons to believe the Christian message and reasons to reject intellectual attacks on Christianity. Its purpose is to help individuals by defeating intellectual objections so they can hear and accept the Christian message.

Apologetics is a thriving business among Protestants; not as much among the Catholics and Orthodox. Presuppositional apologetics, which is controversial even among Protestants and is largely associated with Calvinism, is based on two insights.

The first is that false presuppositions about the basic nature of reality will block an individual’s ability to accept the Christian message. All the correct evidence and reasoning in the world will do him no good until he corrects his false presuppositions.

This is ultimately not a rational process. Calvinism places a special emphasis on the Bible passages teaching that mankind in its natural state cannot help but reject God until God gives the individual spiritual life which is the ability to accept the Christian message. See, e.g., John 6:44 and Ephesians 2:1—10. But God also works through means, and one means of coming to faith is to hear and accept true evidence.

Many Christians reject presuppositionalism because in their experience it looks like circular reasoning: “You cannot prove God, you can only presuppose Him.” Some Christians may talk this way, but you can prove God. You just have to know the correct way. Ultimate truths are not known using ordinary ways of reasoning. See here for more details.

In any case, it’s clear that everyone has presuppositions, that most people are only dimly aware of exactly what they are, and that false presuppositions cause false beliefs, especially false beliefs about the most fundamental facts of reality.

The second basis of presuppositional apologetics is the recognition that mankind has no right to reason in a way that is autonomous from God. There is a truth about God and you have no right to reject it, even if your reasons seem good to you. If you reject a valid proof of God, a presuppositional apologist will not say “OK, let me try a different proof…” He will say “Your thinking is faulty because your presuppositions are false; here’s why…”

Religious discourse between a Christian and a non-Christian is never discussion between neutral parties. Neutrality in one’s stance toward God is impossible. Those who think they are neutrally considering the evidence are unaware of their presuppositions and their commitment to human autonomy. They have taken a side but they have not acknowledged to themselves the nature of this commitment.


These two ideas, that false presuppositions must directly be dealt with and that mankind does not have the right to think any way they want about God seem obvious and important to me, but many serious Christians reject presuppositionalism. Therefore, I propose a thought experiment:

Imagine an individual who considers himself a serious student of mathematics (SSM). His professor is teaching, let’s say, the Fundamental Theorem of Calculus. A rather important theorem. Of Fundamental Importance to Calculus. But SSM rejects it. Maybe he rejects the theorem without studying the proofs offered by the professor and the textbooks. Maybe he studies the proofs but declares them invalid for reasons of his own invention. The professor attempts to reason with him but to no avail. Mr. SSM rejects the theorem and creates his own calculus.

Is SSM justified in his rejection? Obviously not. A neophyte has no right to reject what scholars have learned over hundreds or thousands of years. The Fundamental Theorem of Calculus is the result of thousands of years of mankind’s greatest minds studying mathematics. Since mathematics is real their discoveries are real, and therefore to reject an established truth of mathematics is to be willfully ignorant. You have no right.

[I use “right” in its primitive sense, not its debased modern sense. “You have no right” means “you have no justification.”]

You have no right to reject something true. You have no right to reject the proofs until, at minimum, you understand the subject, what the theorem under discussion says, and the structure and logic of the proofs. And if the theorem is true your reasons for rejecting it don’t matter. You’re wrong.

If the student persists in his rejection of mathematics he should not be treated respectfully by mathematicians. He’s not holding a valid alternative belief. He’s refusing to think correctly. If he persists in conversing with them, mathematicians would be derelict in their duty to the truth if they failed to correct his errors. His false beliefs are ultimately caused by false mathematical presuppositions which must must be identified and corrected. He does not have a right to his own version of mathematics.


If the analogy is not obvious, let’s spell it out. SSM represents a non-Christian. The Fundamental Theorem of Calculus is any truth about Christianity, especially that God exists. The proofs of the Theorem are the evidences for the Christian doctrines. The professor is anyone who presents Christian evidence. SSM’s rejection of the Theorem and its proofs is the typical response of an unbeliever when confronted with Christian doctrine or the evidence which supports it. He’s in the wrong and should be treated as such.

And if it is valid to correct a wayward student’s false mathematical presuppositions and demand that he place himself under the authority of the mathematical truths that mankind has discovered rather than making up his own mathematics, it is even more valid to take a presuppositional approach to conversing with recalcitrant non-Christians.


There are some apparently-valid objections to the analogy. Yes, SSM has made a commitment to mathematics whereas the typical apologist’s interlocutor has no commitment to Christianity. [But he does have a probably-unconscious commitment to his faulty worldview.] Yes, Mathematics is a highly technical subject for the serious study of which most people lack the ability, whereas all people are capable of being serious Christians if they can overcome the impediments. Yes, some creative mathematicians have created their own valid calculi. And there are mathematical theories which are not simply the work of lone crackpots but have gained a significant following, yet are controversial within mathematics. Cantor’s theories of infinity were rejected by the mathematical mainstream when he first propounded them, and there is a minority view of the subject which has followers to this day.

These points are true but irrelevant. The point under discussion is simply this: is it right to emphasize the importance of mankind’s faulty presuppositions and unjustified intellectual autonomy? Yes, it is.


Presuppositionalism is relatively new, at least as a consciously-articulated doctrine. But there is a perfectly natural reason: It was developed in response to the never-before-seen phenomenon of large numbers of ordinary people possessing basically-atheistic worldviews, despite many of them professing to be Christians. Starting in the early 20th Century, and for the first time in the history of mankind, the typical member of a Western nation has basically atheistic presuppositions. For the first time ever, large numbers of ordinary people reject out of hand evidence for Christianity because they believe that miracles are impossible because reality is materialist.

And since most people are unaware of their presuppositions, even when they are pointed out to them, serious Christian apologists need to deal with presuppositions.

Old-school apologetics, which presents evidence without responding in a sophisticated way to worldview-based objections, still suffices for many encounters. And people often change their worldviews spontaneously and unconsciously in response to serious encounters with Jesus Christ and His Gospel. But for most Western non-Christians, false presuppositions and unjustified autonomy block their ability to accept the Christian message. A presuppositionally-aware apologetics has become necessary.

21 thoughts on “Analogy Between Presuppositional Apologetics and Mathematical Proof

  1. Is this an Apologetics System? Is it for formal debate that it has a name?

    I just take where someone’s at and make some arguments. I wouldn’t even know how to categorize them.

    • “I just take where someone’s at and make some arguments.”

      That’s the right approach. But if the other person rejects valid arguments, they have an impediment, and it is helpful to identify it.

      Presuppositionalism is a school (or set of related schools) of Christian apologetics.

  2. I wonder what people could know without religion?
    I mean: think about what you could know, but as well, anyone in history, without referencing religion.

    What would be left over?

    I dont think there would be anything knowable.

    Say, for example, 1000 years ago . What knowledge could there be if I could not compare it to something religious?

    Can you name me any piece of knowledge that doesn’t arise, in some context of religion ?

    • Can you name me any piece of knowledge that doesn’t arise, in some context of religion ?

      Well, yes.

      As Wittgenstein says:

      If a blind man were to ask me “Have you got two hands?” I should not make sure by looking.

      If I were to have any doubt of it, then I don’t know why I should trust my eyes. For why shouldn’t I test my eyes by looking to find out whether I see my two hands? What is to be tested by what?
      (Who decides what stands fast?)

      And what does it mean to say that such and such stands fast?

      • Which is the religion, then? That which I am pointing to in accusation of what is not correct? Or that by which I am deeming correct, yet not calling it religion?
        Which set is informing my philosophy? How am I refraining from involving myself in my own critique?

      • Take the proposition “3733787 is a prime number.”

        It is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is so, because mathematical reality is built that way.

  3. In any argument, a person’s “world-view” (by which I mean one’s network of beliefs and assumptions) is critical, for one’s “world-view” determines one’s methods of doubt and enquiry: on what counts as testing, proving, how one interpret evidence and one’s whole system of verification.

    Wittgenstein gives an example of this:

    “Think of chemical investigations. Lavoisier makes experiments with substances in his laboratory and now he concludes that this and that takes place when there is burning. He does not say that it might happen otherwise another time. He has got hold of a definite world-picture – not of course one that he invented: he learned it as a child. I say world-picture and not hypothesis, because it is the matter-of-course foundation for his research and, as such, also goes unmentioned. ” – On Certainty (Emphasis added)

    Once we formulate the underlying proposition, “substance A always reacts to a substance B in the same way, given the same circumstances,” a moment’s reflection will show that it is neither logically necessary (we can easily imagine it happening otherwise) nor can it be empirically verified, unless we assume that past experience will recur in the future, (which is to beg the question, a pure petitio principii)

    Even in a rigorously deductive science like mathematics, we are familiar with “primitive notions,” for we cannot define all our terms or prove all our propositions without a perpetual regress. Peano argues that, if we are to do arithmetic, we must simply accept the concept of the number zero and the successor function ((f)n= n + 1). To do geometry, Hilbert argues the concepts of point, line, plane, congruence, betweeness, and incidence must be taken as givens.

    Wittgenstein identifies the real philosophical problem here: “The difficulty is to realise the groundlessness of our believing.” (And that applies to our doubting, too)

    • “Once we formulate the underlying proposition, “substance A always reacts to a substance B in the same way, given the same circumstances,” a moment’s reflection will show that it is neither logically necessary (we can easily imagine it happening otherwise) nor can it be empirically verified, unless we assume that past experience will recur in the future, (which is to beg the question, a pure petitio principii)”

      This is where the existence of the God of the Bible comes in. The biblical God created order, and therefore it is a sure inference that order exists everywhere in His creation. Sometimes we misunderstand the order, but we can know order is there.

      Without belief in God, we can have no confidence that chemical substances react in predictable ways. With said belief, we can.

      This is a presuppositional form of argument.

      • An interesting example.

        However, the weakness of the argument is that it is incapable of falsification, so ultimately meaningless. It cannot be used to distinguish any sequence of events from any other.

        Confronted with apparent breaches of “order,” such as accounts of witches flying, tables turning, Saints being levitated, oracles coming true, horoscopes being verified, broken limbs being cured by faith-healing, and the like, its proponents can always claim that there may be some higher law under which such phenomena would come – a pure petitio principii; it assumes that things do work by law, and you haven’t found the law.

        One recalls Mgr Ronald Knox’s treatment of Paley’s Argument from Design: “The argument was a dangerous one, so stated. It took no account of the animal species which have in fact become extinct; it presupposed, also, the fixity of animal types. God’s mercy, doubtless, is over all his works, but we are in no position to apply teleological criticism to its exercise, and to decide on what principle the wart-hog has survived while the dodo has become extinct.”

        Whatever happens, the scientist can simply add new variables to his equations to bring them into line with observation.

      • @Michael Paterson-Seymour

        Regular miracles get inevitably seen as natural phenomena and lose the wonder that comes with its rarity.

        Miracles therefore have to be comparatively rare in order to be seen to have Agency behind its happening.

        The supernatural therefore is the application of agency to phenomena albeit invisible to our immediate observation by Spiritual Agents.

  4. I will enjoy it if a discussion develops around this topic, as I’m temperamentally sympathetic to a (small-“p”) presuppositionalism which encourages humility regarding our knowledge and our methods of knowledge. I’ve always found in (small-“p”) presupp a strong argument for the development of virtue. My experience with more formal presuppositionalist thought is limited but maybe instructive. My youthful Baptist (anti-Reformed) education viewed (much more humbly in my opinion) that presuppositionalism meant the basic reliability of our senses. We were taught no need to argue with modern radical skeptics who denied this reliability because they would always require using the reliability of our senses to refute this very claim. More later upon reading Ayn Rand (I know) her axioms of consciousness, identity, existence – she would never call them presuppositions – interested me in their similarity to much of my early learning. Later I was taught during my associations with RUF campus ministry more big-“P” Presuppositionalism which I ultimately rejected as begging the question for Calvinism. The T and I of TULIP’ry were doing *a lot* of heavy lifting on the big-P view. I remember being told once by an RUF minister that acceptance of Presuppositionalism was a good sign one had fully understood the Gospel. I also was a little annoyed with the “by their fruits” sort of proof offered by Presupp (I felt it was getting things backwards, in an important way). But, Lord, if that hasn’t been hashed out around here recently. None of that is an argument one way or another; just my experiences with differing versions. That said, as a Roman Catholic presuppositionalism is very interesting to me – the imago dei, our yearning for the transcendent, etc. seem very related. I’ll be interested in any further discussion.

    • [p]resuppositionalism meant the basic reliability of our senses

      I should prefer to say the reliability of immediate experience. We learn that chairs exist, that books exist from being told as children to sit on chairs, fetch books &c

  5. “I should prefer to say the reliability of immediate experience.”

    I’ll register your suggestion with the Southern Baptists 🙂

    “ We learn that chairs exist, that books exist from being told as children to sit on chairs, fetch books &c”

    That’s the sort of move – a move from the basic reliability of the senses to how one learns what a chair is – that would have those old timey Baptists praying for you.

  6. landzek,

    Those are very good questions. My experience is that presuppositionalism has (some) rhetorical weight going against atheists. Much less so against various options of theism. Presuppositionalism is essentially Calvinist and has been largely developed in the context of America. And, to me, it shows. Most Big-P Presuppositionalists will ultimately ask you to look at the consequences of the chosen presuppositions as a proof for which presupposed religion is true. Which tends to beg the question for various American parochial Christianities.

    • The classic presuppositionalist argument for Christianity is that all non-Christian wordviews either fail logically, fail to account for important aspects of reality, or else are unliveable by real human beings because they do not fulfill man’s basic nature and needs. This is a subtle form of reasoning which generally requires the subject to be sensitive and perceptive, but I find it persuasive.

    • “Presuppositionalism is essentially Calvinist”
      Interestingly, Blaise Pascal gives a form of it (and Jansenism has often been described as Catholic Calvinism)

      282. We know truth, not only by the reason, but also by the heart, and it is in this last way that we know first principles; and reason, which has no part in it, tries in vain to impugn them. The sceptics, who have only this for their object, labour to no purpose. We know that we do not dream, and, however impossible it is for us to prove it by reason, this inability demonstrates only the weakness of our reason, but not, as they affirm, the uncertainty of all our knowledge. For the knowledge of first principles, as space, time, motion, number, is as sure as any of those which we get from reasoning. And reason must trust this knowledge of the heart and of instinct, and must base every argument on them. The heart senses [Le cœur sent] that there are three dimensions in space and that the numbers are infinite, and reason then shows that there are no two square numbers one of which is double of the other. Principles are intuited, propositions are inferred, all with certainty, though in different ways. – Pensées.

      NB In the 17th century, the meaning of « Le cœur » still retained the Latin sense of “the seat of wisdom, understanding, heart, mind, judgment, &c cf cordatus meaning sensible, possessing common sense.

  7. Alan, presuppositionalism presupposes that the law of non-contradiction is true. So I wonder how you would defend presuppositionalism if you debated it with Prof. Graham Priest. Dr. Priest is a logician who defends dialetheism, the belief that some true propositions have true denials. So he believes that some contradictions are true. I’ll try to cite his article from The Journal of Philosophy, Meanwhile, here’s his article from the Stanford Encyclopedia of Philosophy.

    If you define the law of non-contradiction, LNC, Aristotle’s way, and if that definition is true, the LNC is a metaphysical principle built into reality, not a merely semantic one. After all, Aristotle teaches that nothing can both be and not be in the same respect at the same time.

    If you define the LNC semantically with the words “true” and “false,” you need a way to solve the liar’s paradox. Since I’m sure you’re familiar with it. I’ll describe it briefly for people unfamiliar with it.

    Here are two liar sentences. “I am lying now” and “This proposition is false.” The problem with each of these propositions is that it’s true if and only if it’s false. But the presuppositional apologist I know of assumes the LNC is a semantic principle.

    Here’s a modal question for you, too. How do you deny the consequent in this modus tollens argument?

    If I were a dog, I would bark.
    I wouldn’t bark.
    So I’m a dog.

    Most apologists probably won’t think so technically about the LNC. But if dialetheism is true, that may undermine presuppositional apologetics. Anyhow, I agree with Aristotle’s definition.

    Now, please tell us what the Christian message consists of. Does it go something like this? Jesus Christ is God the Son who became a man to enable us to reconcile with God the Father and go to heaven if they accept Christ as their Lord and Savior. I don’t mean to debate Catholic doctrine with anyone here. No, I ask about the Christian message because interdenominational disagreements makes it hard to tell exactly what message you mean.

    • The Christian message or Good News [εὐαγγέλιον ( euangelion)] is clear enough from the Apostolic preaching [κήρυγμα (kérugma)] contained in Acts

      (1) the age of fulfilment has dawned, the “latter days” foretold by the prophets (Acts 2:16; 3:18, 24);
      (2) this has taken place through the ministry, death, and resurrection of Jesus Christ; (passim)
      (3) by virtue of the resurrection Jesus has been exalted at the right hand of God as Messianic head of the new Israel (Acts 2:33-36; 4:11; 5:31);
      (4) the Holy Spirit in the Church is the sign of Christ’s present power and glory (Acts 2:17-21, 33; 5:32);
      (5) the Messianic Age will reach its consummation in the return of Christ (Acts 3:20; 10:42);
      (6) the preaching of the gospel closes with an appeal for repentance, the offer of forgiveness and of the Holy Spirit, and the promise of salvation (Acts 2:38; 3:19, 25; 4:12; 5:31; 10:43).

      St Paul uses the word εὐαγγέλιον more frequently than any other NT writer. For Paul

      (1)It tells of the sacrificial death and resurrection of Jesus Christ (1 Cor. 15:1-4).
      (2) It is the “gospel of God” because it tells of God’s redemptive work (Rom. 1:1; 15:16; 2 Cor. 11:7; 1 Thes. 2:2, 8-9).
      (3) Since this redemptive work was accomplished in Christ, it is called the “gospel of Christ” (Rom. 1:9; 1 Cor. 9:12; 2 Cor. 2:12; 9:13; 10:14; Gal. 1:7; 1 Thes. 1:8; 3:2).
      (4) Because of what it promises, it is the “gospel of salvation” (Eph. 1:13) or the “gospel of peace” (Eph. 6:15).
      (5) If the gospel was not believed and obeyed, its hearers would be lost (2 Thes. 1:9). If it was received in faith, it became the instrument for one’s salvation (Rom. 1:16).

    • Bill,

      It sounds like you are distinguishing between an ontological law of non-contradiction and an epistemological LNC, with the former being indestructible and the latter being susceptible to certain antinomies. This business of “dialetheism” seems to me to be mostly a word-game rather than a valuable insight. Yes, a sentence like “This proposition is false” doesn’t seem to work, but I don’t see a fundamental problem. You just exclude a few troublesome trick-sentences, and the classical meanings of “true” and “false” retain their usefulness.

      And if you really think the dialetheists are on to something, then I retreat to the ontological PNC and all is well.

      After all, Christian apologetics concerns what is, not just what is said to be true.

      • After all, Christian apologetics concerns what is, not just what is said to be true.

        That reminds me of Moore’s Paradox: “It’s raining, but I don’t believe it’s raining.”

        For Moore, the resolution of the paradox depends upon showing why the assertion that one state of affairs obtains— the state of affairs described by p—implies that another state of affairs—my believing that p—obtains, even though the existence of this second state of affairs is not logically entailed by the existence of the first.

        Wittgenstein, however, writes: “Moore’s paradox can be put like this: the expression ‘I believe that this is the case’ is used like the assertion ‘This is the case”; and yet the hypothesis that I believe that this is the case is not used like the hypothesis that this is the case. (PI p. 190).” This re-formulation of the paradox, directs our attention to the fact that ‘I believe that this is the case’ appears to be used differently in the language-game of asserting and the language-game of supposing.

        He asks, “How did we ever come to use such an expression as ‘I believe …’? Did we at some time become aware of a phenomenon (of belief)? Did we observe ourselves and other people and so discover belief?” (PI p. 190)

        Your distinction between “what is” and “what is said to be true” raises the same problem.

      • Alan, I am distinguishing between an ontological version of the LNC and an epistemological version of it. But I suggest that the epistemological version presupposes the ontological one because to be true, a proposition must conform to reality.

        We need to remember the difference between a truth and a state of affairs because states of affairs make propositions true or false. Our Lord talks ontologically when he says that he is the truth. After all, everyone knows that he’s a person, not a property. So I believe Plantinga reads Platonism into the doctrine about divine simplicity when he thinks it implies that God s a property.

        In his Compendium of Theology, St. Thomas Aquinas argues that the doctrine about the Holy Trinity is logically consisted. Part of his argument goes something like this.

        St. John teaches that the Word was with God and that the Word was God. A word is thought the thinker fathers. God the Father’s Word is his eternally perfect thought about himself. So His thought is exactly like Him. But he’s distinct from that thought. Since the thought is perfect like God the Father that thought, the Word, is also a divine person. Since God the Father loves himself, he also loves God the Son, i.e., the Word, and that love is the Holy Ghost. Maybe that explanation crossed St. Augustine’s mind when he compared the soul to the relations among the three divine persons. Each soul has three faculties: the intellect, the will, and memory. But since the soul is an immaterial object, there’s no way to cut it into thirds.

        That’s why “The Word of God” can denote Christ, the Bible, what God says, and what he wills. When God says “Let there be light” in the first creation account in Genesis, that means he made light exist. After all, why would he speak aloud when no one was there to listen to him? God the Father didn’t need to say that to the other two divine persons since there’s only one divine will that the divine persons have equally. Since they have the same will, they always agree on everything.

        Now let’s return to the nature of truth. We need to reflect on more than epistemology when we wonder about truth. That’s because the truth concept seems circular. Whether you believe the correspondence theory about truth, which I do, the coherence theory, the pragmatic one, or even deflationism, you can still ask whether the theory is true. So again, we’re talking about ore than semantics. We’re talking about what truth is in itself.

        Even if we don’t know what truth consists of, “truth” and falsehood” still signify properties. If there were no God, there would be no truth because there would be nothing at all. That tells me that since there’s a divine mind, there’s would still be truth, even if God’s mind were the only one. Since he causes every effect, each person, place, and thing and the world must correspond to some divine thought or other. So please forgive the metaphorical composition my description implies in this paragraph.

        I think dialetheism is false. And I wonder whether Dr. Priest thought it up because like many other analytic philosophers, his thought is too reductionist. Ayer, Schlick, and Carnap were too reductionist since they rejected metaphysics. Even the Bible presupposes a metaphysics since metaphysics is about being, and if Thomas is right, God is being in itself and a being.


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