It must be established that one does indeed know. There is generally a standard, as in the case of "knowing algebra" that generally applies. If you do pass an algebra test with a perfect score, you can certainly say that you know algebra. However, in saying that you know algebra the knowledge claim doesn't entail knowing with absolute certainty.
What does it mean to know algebra but not know it with absolute certainty? Your example of “knowing” that your car will start is quite different and unhelpful. Possession of a skill and an assumption about the reliability of a car have little in common.
There is no objective standard for what is to count as an objective standard for knowing algebra, that is, for what must be included in an essential skill set. Many students discover this problem when the move from one school to another. They may find themselves way ahead or way behind because the school covers so much more or so much less. The school or school district or state or country may have a standard but it is not a universal standard. Are these different standards all objective? That this and this but not that is included is based on a subjective evaluation of what is essential for knowledge of algebra.
An objective standard can change, it's not necessarily absolute, this of course is seen in Wittgenstein's riverbed analogy.
Or, one might say, there are different standards of what counts as an objective standard. The fact that the standard changes will be seen by some as an indication that it is not objective but a matter of convention. One might, for example, conceive of objectivity as Kant did - universal subjectivity. But this presupposes a fixed architecture of the mind. Hegel emphasizes the importance of history, although he assumed the completion of history. Neither of these concepts of objectivity fit Wittgenstein. This is one reason why Wittgenstein often made appeals to imaginary tribes, with different ways of life and hence different practices, rules, and standards. A tribe may have an “objective” standard for, say, knowing the will of the gods. If one goes out on his own for a period of time without any supplies and returns then he has passed the test and knows the will of the gods. We might see this as evidence of survival skills but not as an objective standard for measuring knowledge of something we think there can be no objective knowledge of since we do not accept the objective existence of gods. They in turn may think we are quite odd for questioning what is for them so obviously evident.
I think there is generally a standard by which we can measure whether or not one knows basic algebra. Whether every school applies the standard is not the point. Much of this has to do with having competent teachers, but I digress. I see your point though about not having an objective standard in the sense that there is no agreed upon criteria that is universally applied. Although certain tests (SAT and ACT) try to measure a certain skill set.
When I say that one can have knowledge of algebra without having absolute knowledge, what I mean is, that one doesn't need to get every problem correct on every test to make the claim that one knows algebra. Absolute knowledge would mean that one would never miss an algebra problem, i.e., I'm 100% certain that I can solve every algebra problem. It's like comparing deductive arguments with inductive arguments. A deductive argument is a proof. Thus, the conclusion follows with absolute necessity from the premises. In an inductive argument one could make the claim that one knows based on what probably follows, i.e., the argument is either strong or weak based on the evidence. The conclusion doesn't follow with absolute necessity, it follows with a certain degree probability. The same could be said of knowing algebra, viz., the claim that one knows algebra isn't a claim that one can get every problem correct. The claim is that I can get many of the problems correct. So if I went to MIT and received a B+ in algebra, I can claim that I know algebra. I know with a high degree of certainty that I can solve a basic algebra problem. I don't know that I can solve every basic algebra problem with absolute certainty.
-- Updated July 8th, 2017, 4:56 am to add the following --
Wittgenstein: On Certainty Post #7
"For it is not as though the proposition "It is so" could be inferred from someone else's utterance: "I know is it so". Nor from the utterance together with its not being a lie.--But can't I infer "It is so" from my own utterance "I know etc."? Yes; and also "There is a hand there follows from the proposition "He knows that there's a hand there". but from his utterance "I know..." it does not follow that he does know it (OC, 13)"
I have already covered some of this, but it should be repeated again and again, until it dawns.
If the skeptic already doubts propositions of the sort Moore is stating, then showing one's hand does absolutely nothing to dismiss these doubts. It is as though the skeptic does not believe the builder when the builder replies that nothing supports bedrock. The skeptic, in this case, is playing a different game. He is playing his own language-game. One has to show the skeptic that he does not understand the language-game that everyone else is playing. However, note that the skeptic has to accept many of the language-games that we accept, even to ask the question. He picks and chooses what he wants to accept and reject to suit his own purposes. The skeptic is confused; however, so is Moore, but not to the same degree. At least we understand Moore - we sympathize with his answers, and so did Wittgenstein. There is something to learn from how Moore replies to the skeptic, and it is important, and it is subtle. The subtlety, however, must be viewed from the angle of On Certainty.
"That he does know takes some shewing.
"It needs to be shewn that no mistake is was possible. Giving the assurance "I know" doesn't suffice. For it is after all only an assurance that I can't be making a mistake, and it needs to be objectively established that I am not making a mistake about that (para. 14, 15)."
According to Wittgenstein knowledge is something that needs to be objectively verified, at least in the majority of cases - if not all. Hence, Moore's example that he knows he has hands, doesn't suffice, because one's assurance is not enough - we need objective evidence. However, isn't Moore giving objective evidence? He says, after all, "These are my hands," and he shows them - what more objective evidence do we need? Is this a good counter-argument to what Wittgenstein is saying?
I think part of the confusion about what Wittgenstein is saying lies in understanding the following point, i.e., do the propositions that Moore lists, normally require such evidence? The answer according to Wittgenstein, is "No." Why? Because according to Wittgenstein these kinds of propositions, or more accurately these basic beliefs, fall outside the language-game of knowing and doubting; which is why applying such terms (knowing and doubting) doesn't work. If you believe, as Wittgenstein seems to believe, that words and propositions are implicitly rule-based, then it follows that you can't use these words in just any context.
If we say "I know...," then this presupposes a justification for "knowing." However, the problem is that these propositions are not the kind of propositions that require a justification, and this seems to be the point.
Another possible confusion is this: Let us assume that there is no language at all; and let us further assume that there are no other people, so all I have are my thoughts. I have thoughts about the world, and about my interactions with the world. There are no propositions, because there is no language. There are no concepts of knowing, doubting, and truth, because these concepts take place in a language, and in the language-game of epistemology. However I can have beliefs, and these beliefs, although not propositionally based, are seen in my actions. I can also have doubts, and these doubts are also observed in my actions.
These beliefs are similar to the example I gave about the dog's belief that it is about to be fed, which is shown by the dog's action of jumping up and down and wagging its tail. Does the dog's belief require propositions or statements? No. They are subjective beliefs based on causal interactions with the world. Does the dog express its subjective certainty that it is about to be fed? Yes. The dog's actions demonstrate this. Is this knowledge? No. Why not? The dog doesn't have the language skills to understand what knowledge is, it doesn't understand the concepts. These concepts occur in language, so if there are no such concepts, then what is the dog doing? The dog is expressing a belief, and that belief is not based on reasons, again, it is causally based. We too, can have such beliefs. In fact, I believe these are the basic beliefs that form what Wittgenstein calls hinge-propositions. They are outside the language-game that Moore is using to describe them. They arise subjectively, and we can express our state of certainty by what we do. We can also do this with doubting
, we can express subjective doubts, and these doubts are also shown in our actions. Let's go back to the dog example, if you continue to pick up the dog's dish without following through with feeding the dog, then the dog will begin to express its subjective doubts by no longer reacting when the dish is picked up. We too can express such doubts by the way we act.