All proofs or arguments (deductive or otherwise) are finite; if they weren't, the conclusion could never be reached. They rely on premises, and those premises, forming the basis for the argument, are unproven.
You could create a proof of those premises; but that new proof would in turn itself rely on unproven premises. So the problem is inescapable. Ultimately, all deductive reasoning depends on premises that aren't deductively proven.
Arguments often also rely on auxiliary premises that aren't explicitly present. They are background assumptions.
Premises fall into three categories: arbitrary premises, like those chosen for an abstract formal system; provisional or working premises; and self-evident truths.
Since arbitrary premises are stipulated, and provisional premises are uncertain, we can move on to the directly apprehended, or self-evident, truth. This could be something experienced, like consciousness, or it could be something relying on intuition or some other form of understanding.
Sometimes, however, things taken to be "obvious" or self-evident are dependent on auxiliary premises that needn't be true, or are otherwise only provisionally true. Provisional truths can be changed to provisional falsehoods (and vice-versa) by additional data, for example.
The problem is complicated by two additional, often overlooked factors: the circularity of definition; and the fact that statements only have truth values if they are sufficiently well-defined.
In order for a statement to be well-defined, each individual term must be well-defined. (Also, of course, the words must be put together in a meaningful way.)
Every dictionary, no matter how technical or comprehensive, contains a finite number of terms. So, every definition, using terms in the dictionary, must point to other terms in the dictionary, which in turn point to other terms in the dictionary. Eventually, some of the definitions must point back, circularly to terms (original or intermediate) that one is trying to define.
So, meaning can't come from from formal definition without circularity. The true meaning comes from personal associations or apprehensions attached to terms. For example, "red" means something to me not because it is defined as a mixture of other colors or as certain wavelengths of light, but because I have seen objects to which the term has been applied.
So, already, the concept of rigor is starting to get fuzzy, or at least subjective.
As for truth values, they can't be assigned to obvious nonsense. "Bling fort shnozzle" isn't true or false, it's simply gibberish.
But what about statements that seem to be talking about something, but involve fuzzy conceptualizations or poorly defined terms, or contradictory elements?
Geometric points are said to have no spatial extension yet somehow occupy and indicate position within space.
Real numbers contain a never-ending series of digits that are nevertheless said to form completed entities in their "entirety".
Deterministic processes are said to be ultimately the result of random quantum events.
"Two things can touch one another." What could be simpler, or clearer?
If there is space between them they aren't touching. And if there isn't space between them they must be spatially coincident, which is to say they must partially occupy the same space simultaneously. Except that because of quantum fuzziness, objects made of atoms with orbiting electrons don't have clear borders. And because of special relativity, simultaneity means something different to observers in different frames. And time is subject to quantum uncertainties. And mathematically precise position ultimately comes down to geometric points, which are nothings pretending to be somethings.