I am reminded of the recent experiment confirming quantum mechanics: reality does not exist until it is measured, but once measured it is a shared reality among the observers[0]. And what is measurement? The assigning of names such that the name can be shared among observers - and once the names are assigned, you can manipulate the names in a more (mathematic) or less (poetic) rigorous fashion (there are, of course, gradations in rigor). So: Anglo-French rationalism is true, at least for pure mathematics which explicitly disclaims application to any specific real-world object - but once we start discussing experience, perception (an accumulation of measurements, again, more or less rigorous) must occur so that reason can be applied. Until each of us has multiple universe-size supercomputers available, mathematical rigor cannot be applied to to the most important aspects of life, such as love (and even then, only if it escapes a Goedel-like undecidability).

[0] https://www.science.org/content/article/reality-doesn-t-exis...