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121 " One important issue that we must address is the possibility that there might be numerous quite different, perhaps inequivalent, algorithms that are responsible for the different modes of mathematical understanding that pertain to different individuals. Indeed, one thing is certainly clear from the start, and that is that even amongst practising mathematicians, different individuals often perceive mathematics in quite different ways from one another. To some, visual images are supremely important, whereas to others, it might be precise logical structure, subtle conceptual argument, or perhaps detailed analytic reasoning, or plain algebraic manipulation. In connection with this, it is worth remarking that, for example, geometrical and analytical thinking are believed to take place largely on opposite sides-right and left, respectively-of the brain. Yet the same mathematical truth may often be perceived in either of these ways. On the algorithmic view, it might seem, at first, that there should be a profound inequivalence between the different mathematical algorithms that each individual might possess. But, despite the very differing images that different mathematicians (or other people) may form in order to understand or to communicate mathematical ideas, a very striking fact about mathematicians' perceptions is that when they finally settle upon what they believe to be unassailably true, mathematicians will not disagree, except in such circumstances when a disagreement can be traced to an actual recognizable (correctable) error in on or the other's reasoning-or possibly to their having differences with respect to a very small number of fundamental issues; "
― Roger Penrose , Shadows of the Mind: A Search for the Missing Science of Consciousness
122 " do not see how natural selection, in itself, can evolve algorithms which could have the kind of conscious judgements of the validity of other algorithms that we seem to have. "
― Roger Penrose , The Emperor's New Mind: Concerning Computers, Minds and the Laws of Physics
123 " In order to decide whether or not an algorithm will actually work, one needs insights, not just another algorithm. "
124 " There is another point that should be made, however, and this is that it need not be the case that human mathematical understanding is in principle as powerful as any oracle machine at all. As noted above, the conclusion G does not necessarily imply that human insight is powerful enough, in principle, to solve each instance of the halting problem. Thus, we need not necessarily conclude that the physical laws that we seek reach, in principle, beyond every computable level of of oracle machine (or even reach the first order). We need only seek something that is not equivalent to any specific oracle machine (including also the zeroth-order machines, which are Turing machines). Physical laws could perhaps lead to something that is just different. "
125 " The foregoing remarks illustrate the fact that the 'tilting' of light cones, i.e. the distortion of causality, due to gravity, is not only a subtle phenomenon, but a real phenomenon, and it cannot be explained away by a residual or 'emergent' property that arises when conglomerations of matter get large enough. Gravity has its own unique character among physical processes, not directly discernible at the level of the forces that are important for fundamental particles, but nevertheless it is there all the time. Nothing in known physics other than gravity can tilt the light cones, so gravity is something that is simply different from all other known forces and physical influences, in this very basic respect. According to classical general relativity theory, there must indeed be an absolutely minute amount of light-cone tilting resulting from the material in the tiniest speck of dust. Even individual electrons must tilt the light cones. But the amount of tilting in such objects is far too ridiculously tiny to have any directly noticeable effect whatsoever. "
126 " is not easy to ascertain what an algorithm actually is, simply by examining its output. "
127 " Moreover, the slightest ‘mutation’ of an algorithm (say a slight change in a Turing machine specification, or in its input tape) would tend to render it totally useless, and it is hard to see how actual improvements in algorithms could ever arise in this random way. (Even deliberate improvements are difficult without ‘meanings’ being available. "
128 " It is a famous theorem first proved by the great (Italian-) French mathematician Joseph L. Lagrange in 1770 that every number is, indeed, the sum of four squares. "
