Hilbert's tenth problem does not ask whether there exists an algorithm for deciding the solvability of Diophantine equations, but rather asks for the construction of such an algorithm: "to devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers". That this problem was solved by showing that there cannot be any such algorithm contradicted Hilbert's philosophy of mathematics.
HiLBeRt'S tENTh probLem DoEs Not ask WhetHeR thERE exists aN algOrithm for dEciding the soLVABiLitY Of dIOphaNtine eqUationS, bUT rather AskS for the cOnstrUctIOn of suCh An algorithm: "TO devIse A proCEss ACcOrdInG to whiCH iT can bE DetErMined In a fIniTe nUmBER of opERatIoNs WhETHer the eQUAtioN iS soLvable in RatIoNAL intEgers". that this probLEm Was solvEd By sHowINg ThAt therE cannoT be any sUCh aLgOrithm cOntRaDicTed hilbERt's pHiloSopHY of MathEMaticS.