## 4 Conclusions  
To summarize the main findings from the recent literature, we can generally argue that although neural network models can exhibit impressive mathematical reasoning skills, their basic abilities for representing and manipulating numbers are not yet fully developed. Even the most advanced deep learning architectures, including large language models, often fail when probed with numerical problems that are posed in a different manner than the training cases or that involve quantities well outside the training distribution. The state-of-the-art performance on challenging benchmarks such as GSM8K is steadily improving, increasing from 58% for PaLM exploiting an external calculator [98] to 78% for Minerva [50] and 92% for GPT-4 [86]. The top accuracy on ASDiv and SVAMP is around 90% for large language models with refined chain-of-thought prompting strategies [99]. However, achieving a high performance on these benchmarks might not be enough to demonstrate numerical understanding, as has also been pointed out by others [100]. Not surprisingly, indeed, the extrapolation capabilities of these gigantic neural networks are still poor: when models are not equipped with external calculators, generalization to unseen (out-of-distribution) numerical ranges requires the discovery of algorithmic procedures. This still constitutes a challenge for deep learning [101] and is indeed considered one of the frontiers in neuro-symbolic research [102]. For example, Minerva achieves over 80% calculation accuracy on 10-digit addition, but this value is never at ceiling even for easy (e.g., 5-digit) problems, and it drops to 20% on 18-digit addition, highlighting that even such a sophisticated model does not master elementary arithmetic. A similar conclusion applies to the extrapolation results presented in [83]; a finer-grained algorithmic prompting strategy allows addition problems to be consistently solved with answers of up to 19 digits in length, even if training examples were limited to 5 digits. However, the accuracy never reaches 100% and after 19 digits, the model runs out of context. The performance of deep learning models is also poor in arithmetic problems specifically designed to test compositionality [103] and it is below the human level in benchmarks that probe approximate numerical knowledge, highlighting that these models still have a limited understanding of quantities and magnitudes [38]. As others advocate, building deep learning systems that can more reliably master basic numerical concepts and simple arithmetic might be a mandatory first step towards successfully tackling complex mathematical reasoning tasks [46]. The perspective offered by decades of research in educational, developmental, and cognitive sciences could provide important insights for the design of such learning systems, especially in terms of the design of training regimens and testing procedures. For example, primary education in mathematics places a strong emphasis on the acquisition of elementary algorithms as the basis for understanding the arithmetic of natural numbers, integers, fractions, and decimals, using a variety of representation formats that exemplify the semantics of the decimal place value system [104]. Importantly, such knowledge is developed on top of the acquisition of number words, numeral systems, and counting procedures [105; 106], which are already fostered during the pre-school period [107]. Furthermore, although language has a central role in this artoulse learning process [108], it is not the only medium through which numerical knowledge is acquired [109]; a key role is also played by the development of basic visuo-spatial skills and geometrical intuitions [110; 111; 112], as well as by the mastery of embodied and material representational systems [63; 113; 114]. Grounding large language models into sensorimotor representations seems to be a promising approach to improve the basic numeracy of AI systems, and the introduction of multi-modal foundation models represent the first step in this direction [115].