Interview: Christoph Benzmueller ... the computer then proved: there exists God (in english)
The next interview with Christoph Benzmueller is about his way to God existence Gödel ontological proof automated proving. Few months ago together with Bruno Woltzenlogel Paleo published an article at arxiv.org about their work.
Could you please, explain the main idea and results of your article?
Kurt Gödel (1906-1978) proposed an argumentation formalism to prove the existence of God. Attempts to prove the existence (or non-existence) of God by means of abstract, ontological arguments are an old tradition in western philosophy. Before Gödel, several well known philosophers, including St. Anselm, Descartes and Leibniz, have presented similar arguments. What motivated Gödel as a logician was the question, whether it is possible to deduce the existence of God from a small number of foundational axioms and definitions, with a mathematically precise and logically formal argumentation chain.
What Bruno Woltzenlogel Paleo from TU Vienna and I did was to formalize Kurt Gödel's argument for God's existence on a computer and to verify it automatically. From Gödel's premises, the computer then proved: there exists God. "Automated theorem proving" is the mathematical discipline where such automated systems are developed. Interestingly, the theorem provers also found some new results about Gödel's argument.
Higher-order Logic :: Higher-order logic is best explained by contrasting it to first-order logic. In first-order logic one can formalize universally quantified statements such as "for all *individuals* holds ... ". However, universally quantified statements such as "for all *properties* holds ..." or "for all *functions* holds ..." are not supported. Higher-order logic in contrast supports all these quantificational statements. In Gödel's ontological proof quantification over properties plays an important role. For a survey on the automation of higher-order logic see:
Formalization :: By formalization we mean the encoding, that is, formulation, of a statement in a formal logic language. Formal logic languages have a well-defined syntax and a mathematically precise semantics. It is always clear what statements formulated in such logic languages actually precisely mean. Examples of formal logic languages include propositional logic, first-order logic, higher-order logic, modal logic, etc. They differ with respect to their expressiveness, that is, with respect to what can actually be expressed with them. Gödel's ontological argument requires higher-order modal logic, which is a very expressive logic language. Evaluation of statements in this logic language has now been automated in our work for the very first time with the help of computer programs, so called automated theorem provers.
In theoretical philosophy, formal logical confrontations with such proofs of God's existence had been so far (mainly) limited to paper and pen. Up to now, the use of computers was prevented, because the "logics" of the available theorem proving systems were not expressive enough to formalize the abstract concepts adequately. Gödel's proof uses, for example, a complex higher-order modal logic to handle concepts such as "possible" and "necessary" and to support quantification over individuals and properties.
Current works of myself and Larry Paulson from Cambridge University, UK, show that many expressive logics, including higher-order modal logics, can be embedded into the classical higher-order logic, which can thus be seen as a universal logic. For this universal logic, efficient automated theorem provers have been developed in recent years, and these systems were now employed in our work.
"From Gödel's premises, the computer
When did you get the idea to work on Gödels ontological proof in "automated way"?
I have been challenged by the idea since about 6 years. It was actually Melvin Fittings book on "Types, Tableaus, and Gödel's God" that initially raised my interest. In the last few years I have then been working on embeddings of various expressive logics in classical higher-order order logics. My main goal thereby was to support improved reasoning technology for various applications, for example, in artificial intelligence. But clearly, this research also paved the way for tackling the ontological argument with computer technology. End of 2012, I then met Bruno Woltzenlogel Paleo in Vienna, where I presented this research to the Kurt Gödel society. Bruno was highly interested in formalizing the ontological argument for personal reasons (he went though some very hard time end of 2012), and he wanted to present a formalization of Gödel's ontological proof as a gift to a priest back home in Brazil. He then contacted me in early 2013 and we decided to jointly work on the formalization and automation of the ontological argument. From the very beginning it was very clear to both of us, that the computer-based formalization and automation of Gödel's ontological argument was of major scientific interest for various reasons. Most importantly, we provide a significant novel contribution towards a computer-supported theoretical philosophy. Remember that such a computational view on philosophy was already pictured by Leibniz.
Is there a story behind creation of the article/code?
Well, yes, and part of the story has already been mentioned above. We succeeded with the first automated proof of Dana Scott's version of Gödel's ontological argument in early July 2013, and it was actually LEO-II, my own higher-order theorem prover, which did prove the result first. Subsequently we reproduced the results with other systems. At the very beginning, I wasn't sure which publishing strategy we should follow. Bruno convinced me that uploading our entire work on GitHub would be the right way to proceed. Raul Rojas at Freie Universität Berlin, whom I had briefly told about our successful work, then prepared the Telepolis/Heise interview with me. And at the same time we did submit the two page article to the arXiv repository in order to have a first reference available. We will, of course, add further publications in the near future; one has already appeared: http://afp.sourceforge.net/entries/GoedelGod.shtml. Others are submitted. These future papers will explain our work and its implications in more depth to different target communities.
What were the reactions after publishing? Were there a lot of them?
Yes, there were strong reactions. In particular, the Telepolis interview and the Spiegel Online article(s) triggered such strong reactions. In fact, the interview on Telepolis was amongst their top ten most read articles in 2013. I assume that a similar statement can be made for some other reports on our work. Many of the reactions to our work were rather irrational, and often they revealed a lack of knowledge in the readers about the nature and role of the ontological argument in philosophy and theology, or they showed a lack of knowledge about formal logic and theorem proving. Other reactions were highly interesting and inspiring. We also got some highly valuable feedback from expert philosophers, and we have started some very fruitful discussions with them. Moreover, we got contacted by a bishop from the roman catholic church, and Bruno met him in Vienna.
"Moreover, we got contacted by a bishop from the roman catholic church, and Bruno met him in Vienna."
Were there also funny or surprising ones?
Well, yes, there are many funny forum comments on the various websites that reported on our work. I also got contacted by several people who presented me their own theories on essential and fundamental questions about the world and about God. Unfortunately, I do not have the resources and the time to comment on all of these requests. Often I have to respond to them that my core interests (so far) have been on the pure logical aspects of the ontological argument.
The project was published at GitHub in open source form. Did many people download (/use) your code? Did somebody modify it in any way? Is there some contribution of community actually?
Unfortunately such statistics data has only recently been made available on GitHub as it seems. This is pity, and I would actually like to know these numbers as well. However, I know from many students at my university that they have downloaded the sources and worked with them. Some students have recently contacted me with some interesting comments and developments. Generally, we would like to invite interested researchers worldwide to join our initiative. Unfortunately, there is still a fair amount of logic training required to fully master the code.
Do you, personally, believe in God? What is your position in this field?
I grew up in a religious family; thus my background can be classified as religious. As a scientist, however, I have developed a purely rational attitude, also regarding fundamental questions as we discuss them here. In fact, my personal belief didn't and doesn't play a role in our work on Gödel's ontological argument. The driving element for me has rather been a deep interest in the logical peculiarities of Gödel's proof and also my growing fascination for the field of metaphysics. But I have to admit that I am in favor of a positive answer to the question of the existence of God. Unfortunately, however, Gödel's proof appears not yet fully convincing to me, since there is some justified criticism regarding some of his axioms. A very interesting question now is whether our technology can eventually help to further analyze and improve the argument.
What topics or subjects are you currently working on and/or planing to work on in the future?
I am currently working on basically four interconnected research directions:
- Six years ago I have started studying to what extent classical higher-order logic can be exploited as a universal logic. The idea is to embed other challenging logics in classical higher-order logics and to apply higher-order automated theorem provers to them. Higher-order modal logic, as required in Gödel's ontological argument, is just one example. Another, even more interesting logic that can be easily embedded is higher-order conditional logic. Also security logics can be automated this way. And most interestingly, combinations of these logics and even parts of their meta-theory can be formalized and automated.
- Together with a group of students I am about to start a re-implementation of my LEO-II theorem prover; I just received some funding from the German National Research Foundation (DFG) to carry out this project. The new system will be called LEO-III. Ideally, I want to optimize LEO-III for the automation of logic embeddings, including the logics as required for automating arguments in theoretical philosophy.
- Together with Bruno Woltzenlogel Paleo I want to formalize a range of variations of the ontological argument. In fact, we would like to formalize major parts of the textbook by Sobel on "Logic and Theism". Moreover, we want to try to further improve Gödel's ontological argument. Currently we try to raise funds for such a project.
- In order to create a larger community we need to significantly improve the user interfaces of our reasoning tools. So far they are suitable only to expert users. Ideally, we should develop a powerful and easy to use reasoning framework for applications in theoretical philosophy. This is another research direction for which I would like to raise funds and create a strong team.