The Mishna in M’nachot 4:1 says that the mitzvah of the white tzizit is not gone if there is no techelet, and the mitzvah of the techelet is not gone if there is no white. That seems a straight forward enough statement.
However, the gemorah's reponse to this statement is essentially a "Are you crazy? How could the Mishna say this?" They then comment that "Obviously, they mean if the tzizit are torn or ripped, it doesn't invalidate your tzizit as long as a "tuft" of the blue is remaining, or vice versa. They then go on to ask what the minimum size of tizit must be before they get turned into tufts.
When Techelet no longer became available during the Geonic period, they obviously said it was ok to have tzizit without techelet.
What was their rationale, i.e. how did they ignore the Gemora, or is there another gemora which contradicts the Gemora in Manachot?
Was the idea of hinuch, so we don't forget about tzizit at all, enough of a reason to possibly violate a Torah commandment of purposefully putting on a four corner garment to put on incomplete tzizit?
The Soncino Translation of the sources: From: http://halakhah.com/pdf/kodoshim/Menachoth.pdf
Our Mishna: Manachot 38a-38b
C H A P T E R I V MISHNAH. THE [ABSENCE OF THE] BLUE [IN THE FRINGES] DOES NOT INVALIDATE THE WHITE, NEITHER DOES THE [ABSENCE OF THE] WHITE INVALIDATE THE BLUE 3 THE [ABSENCE OF THE] HAND-TEFILLAH DOES NOT INVALIDATE THE HEAD-TEFILLAH, NEITHER DOES THE [ABSENCE OF THE] HEAD-TEFILLAHINVALIDATE THE HAND-TEFlllah. 4
Relevant lines from our Gemora:
GEMARA. Must we say that our Mishnah is not in accordance with Rabbi? For it was taught: That ye may look upon it, 5 implies that the [absence of] one invalidates the other. So Rabbi. But the Sages say, The [absence of] one does not invalidate the other. What is the reason for Rabbi's view? — Because the text says, The corner, 6 [which implies that the fringes must be] of the same [colour] as that of the corner, 7 and it also says, A blue thread; 6 and then the Divine Law says. ‘That ye may look upon it’, that is, both must be there together as one. But the Rabbis [say]. ‘That ye may look upon it’, signifies each one by itself. Must we then say that [our Mishnah] is not in accordance with Rabbi? — Rab Judah answered in the name of Rab, You may even say that it follows Rabbi's view, for [our Mishnah deals here] only with the question of precedence. As it was taught: The [proper performance of the] precept is to insert 8 the white threads before the blue; but if a man inserted the blue before the white, it is indeed valid, but he has not fulfilled the precept. What is meant by ‘has not fulfilled the precept’? Talmud - Mas. Menachoth 38b Should you say it means that he has not fulfilled the precept of the white [threads] but has fulfilled the precept of the blue, but according to Rabbi the absence of one invalidates the other! 1 — Rab Judah said in the name of Rab, It means that he has not fulfilled the precept and yet has performed the precept, for ‘has not fulfilled the precept’ only means that he has not performed the precept in the best way. This then explains the clause, NEITHER DOES THE WHITE INVALIDATE THE BLUE; 2 but how can one explain the other clause, THE BLUE DOES NOT INVALIDATE THE WHITE? 3 Moreover, 4 it has been reported: Levi once said to Samuel, Arioch, 5 you are not to sit down 6 until you explain to me the following: THE BLUE DOES NOT INVALIDATE THE WHITE, NEITHER DOES THE WHITE INVALIDATE THE BLUE. What does it mean? — He answered, This refers to the fringes in a [white linen] garment; for it is proper to insert the white threads first, since Holy Writ says ‘the corner’, [signifying that the fringes] of the same [colour] as the corner [must be inserted first]; nevertheless, if one inserted the blue first it does not matter. Well, this explains. NEITHER DOES THE WHITE INVALIDATE THE BLUE, but how can one explain, THE BLUE DOES NOT INVALIDATE THE WHITE? — Rami b. Hama answered, The latter rule refers to a garment that is entirely blue, in which case it is proper to insert the blue threads first, since Holy Writ says ‘the corner’, [signifying that the fringes] of the same [colour] as the corner [must be inserted first]; nevertheless, if one inserted the white threads first it does not matter. Raba objected, Does then the colour affect the law? 7 — Raba therefore explained that [our Mishnah] refers to the curtailment of the threads; thus whether the blue [threads] were curtailed and the white remained or the white were curtailed and the blue remained, it does not matter. As the sons of R. Hiyya said, Curtailed blue threads are valid; curtailed hyssop twigs arevalid. What is the minimum length of a curtailed thread? — Bar Hamduri stated in the name of Samuel, There must be sufficient to make a loop therewith. The question was raised: Does ‘sufficient to make a loop’ mean to make a loop of all the threads together, 8 or of each thread separately? — This remains undecided. R. Ashi raised the question: How is it if [the curtailed threads] are so thick that one cannot make a loop with them, although had they been thinner one could have made a loop with them? — R. Aha the son of Raba answered R. Ashi, They are most certainly [valid], since the precept is all the more noticeable thereby. 9 Who is the Tanna that disagrees with Rabbi? 10 — It is the Tanna of the following Baraitha. For it was taught: R. Isaac says in the name of R. Nathan who said it in the name of R. Jose the Galilean and who in turn said it in the name of R. Johanan b. Nuri, If a man has no blue threads he should insert all white threads. 11 Raba said, You can infer from this 12 that one must make a knot after each joint; 13 for should you hold that this is not necessary, then how could the sons of R. Hiyya have said, Curtailed blue threads are valid, also curtailed hyssop twigs are valid? As soon as the upper knot 14 becomes loose it would all become undone! 15 —
Our Rabbis taught: A linen garment is, according to Beth Shammai, exempt from zizith; 1 but Beth Hillel declare it liable. 2 The halachah is in accordance with Beth Hillel. R. Eliezer son of R. Zadok said, Is it not a fact that any one in Jerusalem who attaches blue threads [to his linen garment] causes amazement? 3 Rabbi said, If that is so, why did they forbid it? 4 Because people are not versed in the law. 5 Raba son of R. Hanan said to Raba, Then let ten people insert it and let them go about in the market place and so the law will be made known to all! 6 People will wonder at it all the more. 7 Then let it be announced at the public lecture! — It is to be feared that people will use imitation blue. 8 But it is no worse than if it were white! 9 —