For the sake of discussion we will assume the teleological argument is sound, as I desire this discussion to focus more on monotheism vs polytheism than atheism vs theism.
So, what is the Teleological Argument for those unfamiliar? It is the watch maker argument, that the universe or certain things in nature appear to be designed and such design demands explanation, that explanation being god(s).
William Lane Craig is a popular proponent of this argument, specifically from the fine-tuning of the universe. His specific argument can be found in his essay “Five Arguments for God” (link below). To copy from his essay, the argument is as follows:
"Here, then, is a simple formulation of a teleological argument based on fine-tuning:
1. The fine-tuning of the universe is due to either physical necessity, chance, or design.
2. It is not due to physical necessity or chance.
3. Therefore, it is due to design"
As we will assume the soundness of the argument for sake of discussion, we now the universe was designed, but does this get us to the monotheistic god that WLC argues for? I think not. There is nothing about the argument that demands the designer be a single entity, it opens in up to the possibility of a multitude of designers working at the fine-tuning of the universe. In fact, the argument itself stems all the way back to Ancient Greece (like many theistic arguments), and thus to polytheists.
A common argument against the polytheistic interpretation is to attempt to argue Law of Parsimony (Occam's Razor), but this, I would argue, is not sufficient. There are two ways in which parsimony works, quantitatively and qualitatively. In regards to theism the qualitative parsimony is based on how many categories or sets of things exist, keeping in mind that sets can be made up of many, one, even no things. Quantitative parsimony deals with the number within a category/set. I think using quantitative parsimony in this instance is flawed, and I will use an analogy to help demonstrate this.
Within physics there is an idea called the One Electron Theory which posits that within the universe exists a single electron. This electron moves both forward and backwards in time, thus creating every instance of an electron and positron in the universe from a single particle. This theory would be favored if we use quantitative parsimony, but the theory is a rejected one. In fact, an even more extreme idea would be to posit that all particles in the universe could, likewise, be a single particle. This idea was proposed by physicist Dr. Weiping Yu in his paper "The U-Theory of Everything (- A single Particle Theory of Universe)".
Both of these ideas would definitely be favorable if we operated the Law of Parsimony quantitatively, but yet physicists reject this and still opt to posit new types of particles one at a time in their models of the universe. This is because of something many that attempt to argue for parsimony forget, syntactic simplicity.
From Stanford:
"Syntactic simplicity, or elegance, measures the number and conciseness of the theory's basic principles"
Essentially, the reason that many particles is better than one is that supposing one particle requires more hypotheses to support the idea while many particles does not. I believe, for a few reasons, that the same can apply to god in the teleological argument.
We can start with the more complex, the fine-tuning, and move on from there. To start with, I believe that it isn't a stretch to think that if there is a single god that there would be a Unified Theory of Everything, which would be representative of the principle in which said monotheistic god designed the world. The problem is that such a theory, while it had its proponents among physicists, has lost popularity.
Stephen Hawking helped popularize the idea of a unified theory of everything but eventually gave up on the notion and declared such a thing to be wishful thinking. The reason for this being Gödel's Incompleteness Theorem. To quote from quantamagazine on :
"He proved that any set of axioms you could posit as a possible foundation for math will inevitably be incomplete; there will always be true facts about numbers that cannot be proved by those axioms. He also showed that no candidate set of axioms can ever prove its own consistency."
A more recent book on the subject is Dr. Lars English's "There Is No Theory of Everything" where he argues that no single set or rules can unify relativity with quantum mechanics as something can be true on a quantum level but false at the gravitational level.
Without a single Unified Theory of Everything, that means the principles behind the design of the universe are many, and we must ask why a single designer would use many principles instead of just one when there exists a Possible World that can be imagined in which a Unified Theory of Everything does exist. I believe that there being one category of beings, god(s), with multiple gods within said set better explains multiple principles behind the design of the universe than one god.
But this only addresses the aspect of the Teleological Argument around the Fine-Tuning of the universe, what about when we get to other aspects of the Teleological Argument?
With how long ago the Teleological Argument was first created it is obvious that it was not always used on the Fine-Tuning of the universe's very principles, like how WLC likes to argue, but about the world and nature itself and what can be found in it. In fact, a well know formulation today is the Watchmaker analogy, but this actually is a modern version of Cicero's argument from his work "On the Nature of the Gods",
"When you see a sundial or a water-clock, you see that it tells the time by design and not by chance. How then can you imagine that the universe as a whole is devoid of purpose and intelligence, when it embraces everything, including these artifacts themselves and their artificers?"
These early formulations did not stem from the in depth mathematics and high level physics of the modern day, however, but of observation of the world itself. This, however, raises a question. If, like how a building demands a builder nature demands a designer, then why is it that societies the world over when observing nature do not conclude monotheism (not even those that originated the argument)?
When we look at history, if the natural world points so easily to a design by god, and more specifically a monotheistic interpretation, then we should expect to find many people from across the world and across time reaching the conclusion of monotheism, but this is not the case. In the Americas, Asia, Africa, etc. we do not see monotheism having arisen but, instead, polytheism. Even monotheistic religions like Judaism can be seen to have henotheistic (and quite possibly polytheistic) roots to them (even Genesis 1:26 uses the plural 'gods'). The historical record would indicate that people from across the world and across time, when looking at the design of the natural world, concluded that there were many gods. The question that must be asked is why.
Why is it that when people see the design of the world around them that they, seemingly independently, came to the conclusion of many designers instead of one? When we account for syntactic simplicity when coming up with a parsimonious answer I believe that we can conclude that it is because there are multiple designers.
Sources:
William Lance Craig's “Five Arguments for God” - https://media.thegospelcoalition.org/ee/articles/Craig_Atheism.pdf
Weiping Yu "The U-Theory of Everything (- A single Particle Theory of Universe)" - https://ui.adsabs.harvard.edu/abs/2014APS..APR.Y9007Y/abstract
Stanford on Simplicity - https://plato.stanford.edu/entries/simplicity/
quantamagazine "How Gödel’s Proof Works" - https://www.quantamagazine.org/how-godels-incompleteness-theorems-work-20200714/