This letter shows that in the mean-field approximation, spin-glasses must be described by an infinite number of order parameters in the framework of replica theory. Spin-glasses are important because they describe one of the simplest cases of amorphous materials. Replica theory is a good framework for studying spin-glasses, as it introduces an order parameter Q_{α,β}, which is the limit of an n×n matrix as n approaches zero. In the mean-field approximation, the statistical expectation value of Q_{α,β} is non-zero only in the spin-glass phase at zero magnetic field. In the Sherrington-Kirkpatrick (S-K) model, the mean-field approximation is exact in the thermodynamic limit. An intriguing feature of this scheme is the necessity of reaching the limit n=0 as analytic continuation in n from positive integers. In the standard treatment of the S-K model, Q_{α,β} is assumed to be equal to q, the Edwards-Anderson order parameter. However, this assumption leads to results that contradict computer simulations of the S-K model and results in a negative entropy at zero temperature, which is not possible. It has been suggested that the true value of Q_{α,β} is not symmetric under permutations of the indices, implying that replica symmetry is broken. Various patterns of symmetry breaking have been proposed, and it has been noticed that the pattern of symmetry breaking depends on a continuous variable which must be treated as a variational parameter. If the matrix Q_{α,β} is parametrized as a function of three variables, good results have been obtained for the S-K model. The agreement with computer simulations is excellent, and the zero-temperature entropy is small. If we generalize this approach, the matrix Q_{α,β} becomes a function of many parameters, with the three-variable case being only the first step toward this direction. The matrix Q_{α,β} belongs to a zero-dimensional space, so it is not evident how to write down the general matrix of this space. The only known procedure involves using a simple ansatz for integer n, which are analytically continued to n=0. The matrix Q_{α,β} depends on five parameters, and for different values of m1 and m2, different results are obtained. The free energy F(Q) can be obtained by substituting the ansatz into the expression for F(Q), and in the limit n→0, the free energy is given by a specific expression involving the parameters q_i and m_i. The results show that the free energy F(τ) is given by a cubic and quartic term in τ, with coefficients that depend on N. For N=1, the results agree with the predictions of Thouless, Anderson, and Palmer for temperatures greater than 0.2. For N=2, the results agree with the TAP predictions for temperatures greater than 0.1. TheThis letter shows that in the mean-field approximation, spin-glasses must be described by an infinite number of order parameters in the framework of replica theory. Spin-glasses are important because they describe one of the simplest cases of amorphous materials. Replica theory is a good framework for studying spin-glasses, as it introduces an order parameter Q_{α,β}, which is the limit of an n×n matrix as n approaches zero. In the mean-field approximation, the statistical expectation value of Q_{α,β} is non-zero only in the spin-glass phase at zero magnetic field. In the Sherrington-Kirkpatrick (S-K) model, the mean-field approximation is exact in the thermodynamic limit. An intriguing feature of this scheme is the necessity of reaching the limit n=0 as analytic continuation in n from positive integers. In the standard treatment of the S-K model, Q_{α,β} is assumed to be equal to q, the Edwards-Anderson order parameter. However, this assumption leads to results that contradict computer simulations of the S-K model and results in a negative entropy at zero temperature, which is not possible. It has been suggested that the true value of Q_{α,β} is not symmetric under permutations of the indices, implying that replica symmetry is broken. Various patterns of symmetry breaking have been proposed, and it has been noticed that the pattern of symmetry breaking depends on a continuous variable which must be treated as a variational parameter. If the matrix Q_{α,β} is parametrized as a function of three variables, good results have been obtained for the S-K model. The agreement with computer simulations is excellent, and the zero-temperature entropy is small. If we generalize this approach, the matrix Q_{α,β} becomes a function of many parameters, with the three-variable case being only the first step toward this direction. The matrix Q_{α,β} belongs to a zero-dimensional space, so it is not evident how to write down the general matrix of this space. The only known procedure involves using a simple ansatz for integer n, which are analytically continued to n=0. The matrix Q_{α,β} depends on five parameters, and for different values of m1 and m2, different results are obtained. The free energy F(Q) can be obtained by substituting the ansatz into the expression for F(Q), and in the limit n→0, the free energy is given by a specific expression involving the parameters q_i and m_i. The results show that the free energy F(τ) is given by a cubic and quartic term in τ, with coefficients that depend on N. For N=1, the results agree with the predictions of Thouless, Anderson, and Palmer for temperatures greater than 0.2. For N=2, the results agree with the TAP predictions for temperatures greater than 0.1. The