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Although, I can conceive of a potentially infinite amount of time ahead of me, for it is not actual as it remains to be experienced, I can't conceive of a potentially infinite amount of time behind me, for having been experienced it is actual, and being actual it is finite, as I can't conceive of actual infinities.
Hence there is a beginning to time.
Has anyone discussed this argument in the literature?
There is a pretty fundamental fallacy at play in the argument. Let's take it step by step.
I can conceive of a potentially infinite amount of time ahead of me, for it is not actual as it remains to be experienced
True. Naturally, since no one (yet) has proven to be immortal, the vast majority of that infinite time "ahead of you" will also be "without you"; but either way, we can conceive of, say, the year 2090, or the year 20000000[...insert zeroes at leisure...]000090.
I can't conceive of a potentially infinite amount of time behind me, for having been experienced it is actual, and being actual it is finite,
Wrong. You have experienced a small fraction of that potentially infinite time "behind you", but there is nothing stopping you from conceiving of, say, the year 1890, which (to all appearances) occurred without your presence. Of course, if you are a Buddhist, the year 1890 did include your presence, in some manner or another, but that's neither here nor there-- and that's precisely the flaw in the argument. Imaginary time is symmetrical, whether we imagine ourselves around to experience it or not.
Has anyone discussed this argument in the literature?
Nope, for reasons that should be apparent.
The question seems to invoke a rather specious argument but it seems like a genuine concern so I'll counter it with another thought experiment: Why can't you just always conceive of an earlier time? Let's say you're right, and that time started at time T. Certainly one can conceive of a time just before T, right? In fact it seems to be embedded in our very concept of time that there is always a before and after...
Imagine time as circular. That would mean it has neither a beginning nor an end, yet it is still finite in length.
Time has traditionally been viewed as either like a circle or like a line. Plato, Aristotle and many other Greek and Roman thinkers, particularly the Stoics, advocated a circular view of time. Linear time first appeared in Hebrew and Zoroastrian Iranian writings. Seneca was an advocate of linear time. Augustine thought time was specifically like a line segment. It had a distinct beginning and end, from Genesis to judgement day. Later on Aquinas agreed, and even further on Newton mathematically represented time as a line in his equations. Prominent thinkers such as Barrow, Leibniz, Locke and Kant all agreed with a linear type of time, and in the 19th century time was widely regarded, in both philosophy and science, like a line. It wasn't until 1949, when Kurt Godel, working with Einstein's equations, developed "closed loops of proper time", which are semi-circular in that they allow one to end up where they started after going forward in time.
I accept the result of Kant's antinomy on time, from which he deduces that time is a human interpretation, imposed on thought, and not a real thing. Then to address a question like this, we have to decide what we are really talking about when we speak of time passing. For me, that is the Second Law of Thermodynamics.
If time is not a dimension, but is the effect of the ongoing increase of entropy, then it cannot be infinite in either direction, especially if it is linear.
As we reach utter pattern-lessness, time should have more and more difficulty moving forward. If there is in fact a state of such disorder that it no longer allows for the osmosis of heat, time will eventually stop. At the same time, if entropy decreases backward in time, it must lead back to a state of very high order. At either end, time slows asymptotically as either freedom or movement becomes so rare that what is going on cannot be seen as the passage of time.
In one interpretation, the fluctuation theorem says time can flow in either direction, but must reverse at a point of zero or maximal entropy. So if we managed at the right point to reverse the dependency of our processing and storage mechanisms, we could experience the time reversal as continuing forward, with all of our thermodynamics reversed.
This implies to me that the two ends of time need to be symmetrical, and it cannot be possible to experience a limitless future with a fixed beginning, in a consistent way. If time is reversible, imaginary time can be boundless in either direction, and real time as process we actually observe must stop at both ends.
... as I can't conceive of actual infinities. Hence there is a beginning to time. Has anyone discussed this argument in the literature?
Yes, actually even someone who could conceive of actual infinities, namely Georg Cantor. Here are some quotes of his. They have been stated in private letters but meanwhile exist in printed form:
"With respect to the creation of the world and its temporal beginning I completely agree with you Reverend Father but I also agree with St Thomas Aq., who contests in his Opusc. de aeternitate mundi the mathematical provability of this theorem (that a temporal beginning of the world has to be assumed). [...] If it is said here that a mathematical proof of the beginning of the world in finite time cannot be given, then the emphasis is on the word 'mathematical' and only in that respect my opinion is in agreement with St Thomas. On the other hand, just based upon the true teaching of the transfinite, a mixed mathematical metaphysical proof of the theorem might well be possible. In so far I differ from St Thomas, who holds the opinion: 'Only by belief we know that the universe did not always exist, and that cannot be checked by proof on its genuineness'." [G. Cantor, letter to A. Schmid (26 March 1887)]
"I definitely agree with you, Reverend Father, in the assumption of a temporal beginning of the world. I have always considered the contrary dogma of present natural sciences as violating good reason in highest degree." [G. Cantor, letter to A. Schmidt (5 Aug 1887)]
"I do not only maintain with all Christian philosophers the temporal beginning of the creation, I also claim like you that this truth can be proven by rational reasons. [...] The foundation of actually infinitely great or, as I call them, transfinite numbers does not entail that we have to refrain from rational proofs of the beginning of the world." [G. Cantor, letter to J. Hontheim (21 Dec 1893)]
"[...] for instance, the time elapsed since the beginning of the world, which, measured in some time-unit, for instance a year, is finite in every moment, but always growing beyond all finite limits, without ever becoming really infinitely large." [G. Cantor, letter to I. Jeiler (13 Oct 1895)]
"With great interest I have studied your essay: 'The teachings of holy Thomas of Aquino about the possibility of a creation without beginning.' It was very satisfying for me to see the position of holy Thomas concerning actual infinity be discussed from such a profound expert and to learn that I had correctly understood holy Thomas in this point and related questions, in particular that his arguments against the actual infinite in creatis or against the possibility of actually infinitely great numbers has, for himself, not the meaning of a demonstratio, quae usquequaque de necessitate concludit leading to metaphysical certainty, but was in his own eyes only probable to a certain degree." [G. Cantor, letter to T. Esser (5 Dec 1895)]
Time is infinite - there was no beginning - there couldn't be, just like numbers you can subtract 1 from any number, infinitely. If you believe there was a year ago, and a 10 years ago, then theres no reason you should have to stop adding zeros, infinitely, to represent a time that happened.
