Recently, a fairly popular YouTube atheist made a video that responded to some snippets of what I said regarding axioms.[acp footnote]1[/acp] Particularly, the atheist, Gary Edwards, attempted to refute what I stated regarding the nature of axioms. He did not directly address the axiom of revelation, but instead he dismissed it with an expletive. Unfortunately, the video consisted mostly of mockery. Mockery is a common theme with individuals that are insecure with the arguments that they are putting forth. If Edwards had left all of the mockery out, his video could have been under two minutes. Also, omitting mockery would avoid potential embarrassment after being corrected on the nature of axioms.
Edwards set out to, as he insinuated, to use philosophy to refute my claims. Using philosophy is fine, but the question is, is Edwards using good philosophy or bad philosophy? Unfortunately, it appears that Edwards doesn’t know as much about the nature of axioms as he thinks he does. Since his video focuses on the nature of axioms, that is what this blog post will address.
It should also be noted that a lot of atheists were in full support of his video. Considering that atheists commonly accuse Christians of not intellectually examining what apologists says, Edwards should be quite concerned that none of the atheists seemed to notice that he does not understand the nature of axioms. Perhaps the atheists should be referred to the idiom, “That’s the pot calling the kettle black.”
Every worldview starts with an axiom known as a first principle. A first principle is an indemonstrable axiom. While there may be evidence for some axioms, evidence is not the same as demonstration. In fact, evidences are viewed in light of one’s worldview. Therefore, if evidence is used in an attempt to “confirm” your first principle, then the evidence is a person’s first principle, not the first principle that the person is trying to confirm. Axioms can also be used due to their intrinsic merit.(Self-Authentication)[acp footnote]2[/acp]
There is, of course, an issue that surfaced at the beginning of Edwards’ video. The video started with a clip of me discussing first principles. Edwards then cut himself in and insinuated that he wanted to talk about first principles. Unfortunately, Edwards then gave an example of a mathematical axiom in Euclidean Geometry. Of course, this would not suffice as a first principle in the context of epistemology, which is what I was discussing in the video that he took a cut from. I doubt that Edwards would posit any axiom of Euclidean Geometry as a first principle in an effort to answer the problem of skepticism. So, at the start of the video, Edwards begins by committing a category error, for the axiom he used to try and refute my statements regarding first principles had nothing to do with epistemology.
What will follow are quotations from the video along with my responses.
“So your axiom, your self-evident truth, your uncontroversial riverbed proposition, is:
[r] “The bible is the word of God”
It would appear that Edwards doesn’t have a good understanding of the nature of axioms. Some argue that axioms must be self evident. Of course, I have questions about what that means. Does it mean that to deny the axiom would lead to absurdity? If so, then the axiom of revelation would be considered self evident, for its denial would lead to skepticism. Does he mean that the axiom must be indisputable? If so, how is it determined that the axiom is indisputable? Why does the axiom have to be indisputable when we know that majority consensus is not a logical justification for truth? Nevertheless, being self evident and indisputable is not a necessary qualification for axioms. Some axioms are held to due to their intrinsic value.[acp footnote]2[/acp] Shouldn’t axioms be weighed by how well the axioms are able to solve the problems that are posited in philosophy? It is quite strange, but still unsurprising, that Christianity is not afforded the same courtesy as secular ideas.
“So hamster-faced dogma nipple, Jason Petersen seems to think that we can pull controversial claims out of our posteriors; whilst confidently asserting that they’re axioms.”
Edwards is referring to the axiom of revelation: The Bible is the Word of God. Technically, anyone can start with any axiom they’d like to assert. The problem is, of course, that any other axiom other than scripture will leave a worldview in skepticism. Every worldview must answer the question: How can things be known? If the axiom cannot answer this question, then the worldview in which the axiom is predicated upon falls. After all, if knowledge can’t be justified then all views on meaning in life, metaphysics, science, etc. are all meaningless, for knowledge can’t be justified. I had one atheist the other day say, “Okay, fine. My axiom is that Christianity is false.” I then asked him how that helped him solve the problem of skepticism. Of course, he said that it didn’t help him. I then pointed out that if his first principle, Christianity is false, can’t deal with the problem of skepticism, then he is left in skepticism. Thus, he can’t even be sure that Christianity is false.
“Thats just bull****. Axiom’s don’t have to be demonstrated; but that doesn’t mean they can’t be. For example, this is a diagrammatic representation of a Euclidean axiom after Tarki.”
Is it? Axioms that can be demonstrated are not axioms. I can think of no axiom that can be defended without appealing to the axiom itself. If Edwards is aware of one, I would like him to give an example. Any axiom that can be demonstrated by something independent of the axiom is not the axiom, rather, what he uses to “demonstrate” the axiom is preceding the axiom. This is, of course, out of order. Axioms are starting points, not conclusions.
Edwards also seems to be unaware of how axioms in mathematics works. Any demonstration of the axiom would have to assume that the axiom is true. The axioms are ultimately indemonstrable, but mathematicians test axioms for logical consistency. Of course, I doubt Edwards would take my word for it. Perhaps he will take a mathematician, Dr. Robert G Brown, from Duke University’s word for it[acp footnote]3[/acp]:
Axioms are not self-evident truths in any sort of rational system, they are unprovable assumptions whose truth or falsehood should always be mentally prefaced with an implicit “If we assume that…”. Remembering that ultimately “assume” can make an a** out of uand me, as my wife (a physician, which is a very empirical and untrusting profession) is wont to say. They are really just assertions or propositions to which we give a special primal status and exempt from the necessity of independent proof.
It appears that Edwards doesn’t know quite as much about mathematical axioms as he attempted to insinuate in his video.
It demonstrates, for those familiar with the form, that for any given angle, and any interior point v, there is a line segment, inclusive of v, with endpoints each side of the angle. Thereby doing what Petersen claims cannot be done. Axioms are indeed often ‘primitive’ or ‘basic’. Which means we need not offer any further explanation of them in terms other than those in which they are formulated. As with the diagrammatic example. But that doesn’t mean that they cannot be demonstrated — not even in their own terms.
To be clear, circular reasoning is not problematic when appealing to a first principle or axiom, but since Edwards is making an argument that the axiom is demonstrable, any appeal to circular reasoning becomes problematic.
Edwards asserts that the axiom is demonstrated in this diagram. However, the diagram assumes that the axiom is true. The diagram doesn’t demonstrate the axiom, rather, it draws what the result would look like if the axiom were true. This is not a case of demonstration, rather, Edwards is attempting to demonstrate the axiom by assuming that the axiom is true. It is a shame that Edwards overlooked this elementary fact. The diagram, of course, is self consistent, but there are also consistent non-Euclidean geometries that reject portions of Euclidean Geometry. It is unclear whether Edwards is ignorant of this fact, or if he just left it out.
There are, of course, other issues with Edwards’ “demonstration” of this axiom. His statement was, “It demonstrates, for those familiar with the form, that for any given angle, and any interior point v, there is a line segment, inclusive of v, with endpoints each side of the angle”(Emphasis added). So, the statement says is a universal one. The problem is, the diagram is not universal. The diagram assumes the truth of the axiom, and only gives a few examples. To say that this axiom holds true for all angles would go beyond the bonds of what the diagram supposedly demonstrates. The diagram certainly is not a diagram that demonstrates that any angle will have endpoints at each side of the angle in regards to V. In other words, the justification that is given through the diagram is an attempt at induction. He attempted to take a few visual examples that presupposes the truth of the axiom, and then made an inductive inference for all given angles. In the context of epistemology, several other assumptions prior to the axiom must be assumed:
1. The law of contradiction: If the law of contradiction is false, then there can be no differentiation concerning different angles in Euclidean Geometry.
2. Induction: Edwards made a generalization in an attempt to “demonstrate” his axiom.
3. Sense perception: Edwards must utilize the five senses to perceive the diagram.
It is known among mathematicians that Euclid proved his theorems by contradiction. In order for Edwards to do the same, he must give a justification for the prescriptive law of contradiction that comports with his own worldview. Edwards must also solve the problem of induction, since his attempted demonstration of the axiom is predicated upon inductive reasoning. And last, he must be able to give justification concerning the validity of sense perception in accordance with his worldview. Unless he can solve all three of these issues, he can have no rational justification for accepting Euclid’s axioms in geometry.
Axioms are indeed often ‘primitive’ or ‘basic’. Which means we need not offer any further explanation of them in terms other than those in which they are formulated. As with the diagrammatic example. But that doesn’t mean that they cannot be demonstrated — not even in their own terms.
Unfortunately, the diagram that Edwards provides is predicated on the assumption that the axiom is true. This is, of course, the approach that anyone should take when defending an axiom. For instance, scripture is axiomatic for the Christian, and the Christian will use scripture when arguing for scripture. If Edwards has no problem with assuming the axiom when defending it, then it would seem that if he wants to reject the axiom of revelation, he must give up Euclidean Geometry, for the method of justification of the axioms are predicated on the axioms themselves. It is also important to note that the diagram that Edwards used was not discovered, rather, it was selected. It was, of course, selected because the diagram assumes that the axiom is true.
I suppose that Edwards has two choices:
1.) He can stop complaining about the axiom of revelation.
2.) He can give up Euclidean Geometry.
Now, on with the rest of his video:
So, ever the consummate conceptual proctologist, Petersen has just pulled out the bare generalisation that axioms can never be demonstrated. In order, it would seem, to avoid the perfectly sensible objection that:
“The bible is the word of God”
…is not even evident in its own terms. On pain of that very type of circularity cited in the most introductory of logic-books (“…because the bible says so”).
Unfortunately, it appears that Edwards has not realized that his justification for an Euclidean Axiom(I suspected that he may have been referring to the parallel postulate, but I asked him for further clarification. He said he is referring to Pasch’s axiom.) is circular. It seems that Edwards is unaware that axioms are not only are indemonstrable, particularly in mathematics and logic, but any defense of mathematical axioms will assume the truth of the axiom that is being defended. This was illustrated perfectly when he appealed to a diagram to “demonstrate” Pasch’s axiom. Mathematical axioms are tested for internal consistency, but they cannot be proven.
In the same way, in order to argue for the truth of scripture, we are going to appeal to scripture. This is no different of an approach than Euclid’s axioms or Pasch’s axiom. After all, most philosophers, such as Kant, have appealed to intuition to justify the use of Euclid’s axioms. Of course, intuition is not demonstration.
Ironically, in the next section of my response, we will see that Edwards says that I failed to put an argument forth. Of course, the axiom of revelation is not an argument, rather, it’s a first principle. So, it appears that Edwards either ignorantly rejects axioms on the basis of circularity, although he will accept Pasch’s axiom on the basis of circularity, or he doesn’t understand the difference between a first principle and an argument. Since Edwards did not quote me putting forth an argument in that video, circular reasoning is not a valid criticism if he wants to maintain that he accepts the principles of Euclidean Geometry.
“Where Peterson fails calamitously, of course, is in his violation of the cardinal rule of presup. He broke the circle-jerk in order to try and make an argument. At that point the pooch was screwed. Because — as even perpetual YouTube pooch G-man knows — proper presup is supposed to go like this:
It’s either God or absurdity.”
It appears that Edwards hasn’t read authors such as Van Til or Gordon Clark. My approach is no different than the dilemma of God or absurdity. I simply argue that if one does not start with scripture, he will end up in skepticism. Skepticism, of course, is absurd.
I recommend that Edward study a position he plans to criticize in order to avoid making ill-informed criticisms of a subject matter. I suspect, based on Edwards’ videos and articles, that the only method of presuppositional apologetics that he is aware of is Sye Ten Bruggencate’s method. Apparently he is unaware of those like Van Til and Greg Bahnsen(Sye derived his method from some principles that Greg Bahnsen taught.) used an argument known as the Transcendental Argument for God. Gordon Clark, on the other hand, did not put forth a specific argument. Rather, he said that all worldviews that start with a first principle other than scripture would not survive an internal critique and would be reduced to skepticism.
Must Axioms be Demonstrated?
This is an odd question, but it is a relevant question. Must all claims be demonstrable? If so, would this include axioms? Postulates and theorems are to be demonstrated, but if axioms are also to be demonstrated, then it would lead to the conclusion that all claims, be they axioms, theorems, propositions, or postulates, must be demonstrated, then it would lead to an infinite regress of justification. A worldview or model that falls into an infinite regress of justification falls right into the problem of skepticism. Certainly, no one is able to give an infinite amount of justifications for a single proposition, axiom, theorem, or postulate.
It appears that Euclidean Geometry falls into this trap. In Euclid’s writings, Euclid started with axioms. He deduced theorems and postulates from those axioms, and then attempted to prove the axioms using the theorems and postulates. This of course, would be circular reasoning. Once one sets out to argue that an axiom can be proven or demonstrated, but starts with the axiom to deduce the theorems and postulates that prove the axiom, the one that is arguing for the axiom is begging the question. If Edwards or Euclid wishes to argue that the axioms of Euclidean Geometry or even Pasch’s axiom are proven by the theorems and postulates that are deduced from the axiom, then they have effectively murdered logic by arguing that circular reasoning is a valid form of argument. This is, of course, problematic because Euclidean attempted to prove theorems through contradiction. If logic is dead, then contradiction cannot be used as a method of proof.
There is, of course, a question that arises from Euclid’s approach. If Euclid’s axioms can be demonstrated through the theorems and postulates that are deduced from his axioms, then where is it that Euclid is actually starting? Is he saying that his axioms are justifiable because they are proven by the theorems and postulates that are derived from that axiom? If so, it would appear that he is starting with the theorems and postulates, not the axioms. This would, of course, turn the axioms into conclusions and the theorems and postulates into axioms. This is, however, doing philosophy backwards. One does not start at theorems and postulates and then draw the conclusion that an axiom is true. Rather, the axiom must be assumed, and to reiterate, to argue that the axiom is demonstrated by theorems and postulates that are derived from that axiom is an argument that results in circular reasoning. This would, of course, render the argument for the demonstration of the axiom logically invalid. So then, is Euclid starting with the axioms or the theorems and postulates? If there is no starting point, then the the mathematical model cannot get off the ground. How can one say they start at the beginning with a first principle when they attempt to say that other principles justify the first principle? If there is a preceding justification for saying that an axiom is demonstrable, then the person that is arguing for the axiom is not starting at the axiom they are arguing for.
It is unclear if Edwards thinks that ALL axioms must be demonstrable, but if he thinks that there are some axioms that need no demonstration, then one must ask how he can hold that scripture needs to be demonstrated whereas other axioms do not need to be demonstrated. This would, of course, uncover a prejudice that is fueled by his atheism. To simply assert that scripture must be demonstrable even if it’s axiomatic, Edwards would have to reject all axioms in order to be consistent with his objection.
Gordon Clark points out this problem as well[acp footnote]4[/acp]:
The demonstration of a proposition, such as any theorem in geometry, is completed only when it is referred to the axioms. If the axioms in turn required demonstration, the demonstration of the proposition with which we began would remain incomplete, at least until the axioms could be demonstrated. But if the axioms rest on prior principles, and if these too must be demonstrated – on the assumption that every proposition requires demonstration – the proof of our original theorem would never be finished. This means that it would be impossible to demonstrate anything, for all demonstration depends on indemonstrable first principles. Every type of philosophy must make some original assumptions. And if the law of contradiction is not satisfactory, at least these Heracliteans fails to state what principle they regard as more so. Nonetheless, though the law of contradiction is immediately evident and is not subject to demonstration, there is a negative or elenctic argument that will reduce the opponent to silence.
A Question for Edwards
Edwards insinuated at the beginning of his video that he wanted to talk about first principles. Given that he brought up Pasch’s axiom, is Pasch’s axiom Edwards’ first principle? If so, how would Pasch’s axiom solve the problem of skepticism in the context of epistemology? If Pasch’s axiom is not Edwards’ first principle, then what is the first principle in Edwards’ worldview that he would consider to be better than the axiom of revelation?
Edward’s video had nothing to do with epistemology, rather, it had to do more with mathematical axioms. Of course, one cannot have mathematics without being able to answer the problem of skepticism. That being said, it appears that Edwards’ response was not even within the context of the subject I was discussing when I talked about first principles. Most of Edwards’ video consisted of mocking me and other apologists, but at least the video was able to bring to light some important issues that need to be addressed. One of them, of course, being that Edwards’ understanding of axioms is not quite as prolific as his ego.
4. Thales to Dewey, 2000, pg. 88