Aztec mathematics, a sophisticated system developed by the Mesoamerican peoples, played a critical role in the culture, governance, and daily lives of the Aztec civilization. This mathematical system was characterized by its unique numeral framework and its intricate calendrical systems, which were deeply intertwined with the belief systems, agricultural practices, and social organization of the Aztec people. Unlike the base-10 system prevalent in much of the world today, the Aztec numerical system was based on a vigesimal (base-20) framework, heavily influenced by their agricultural lifestyle and cosmology.

At the heart of Aztec mathematics was its numeral system, which used a variety of symbols to represent different values. The basic symbols of the Aztec numeral system included dots, which represented units; horizontal lines, representing fives; and a shell symbol, which stood for the number 20. This system extended to larger numbers through combinations of these base symbols. For instance, to represent the number 23, the Aztecs would display one shell for 20, three dots for three additional units, demonstrating their innovative method of numeration.

This vigesimal system allowed the Aztecs to perform complex calculations and maintain detailed records for various purposes, including taxation, tribute collection, and trade. Each symbol used in the Aztec numeral system contributed to a comprehensive understanding of trade goods and resources, which were often represented in terms of agricultural produce like maize and beans. The recording of such information was vital for the functioning of the empire and was often inscribed onto codices, illustrated manuscripts that documented a wide array of subjects, including mathematics, history, and mythology.

In addition to their numeral system, the Aztecs developed an elaborate calendrical system that reflected their agricultural cycle and religious beliefs. This system consisted of two main calendars working concurrently: the Tonalpohualli, a 260-day ceremonial calendar, and the Xiuhpohualli, a 365-day solar calendar. The Tonalpohualli was intricately connected to the agricultural cycles, with each day associated with different deities and rituals intended to garner favor for their crops. The 260 days were divided into 20 periods of 13 days, each with its unique deity and significance, serving as a framework for divination, ritual activities, and social organization.

The Xiuhpohualli, on the other hand, was primarily concerned with the solar year and agricultural cycles. This calendar consisted of 18 months of 20 days each, with an additional short month of 5 'nameless' days, culminating in a total of 365 days. Each month was marked by a specific agricultural activity, such as planting or harvesting. The synchronization of these two calendars was crucial for the Aztecs, as it allowed them to align their spiritual practices with agricultural and seasonal requirements, thereby reinforcing their connection to the natural world.

Central to the calendrical systems was the significance of time in Aztec culture. The Aztecs viewed time as cyclical rather than linear, which influenced their understanding of history, prophecy, and the rituals necessary for maintaining cosmic order. Each day in the Tonalpohualli had a distinct meaning and associated deities, which the Aztecs believed influenced human affairs. Therefore, precise calendrical knowledge was not merely practical; it was also spiritual, impacting decisions about war, agriculture, and daily life.

Aztec mathematics did not only serve religious and agricultural purposes but also had significant implications for their architectural and astronomical achievements. The sophistication of their calendar systems is evident in their monumental architecture, which was often aligned with celestial events like the solstices and equinoxes. The Templo Mayor, the main temple of Tenochtitlan, was built at a site and with a design that reflected their advanced understanding of astronomy and mathematics. The alignment of structures and the timing of religious ceremonies showcased the unity of mathematics, spirituality, and governance in Aztec society.

In terms of astronomy, the Aztecs devised an intricate system to track the movements of celestial bodies. Their observations of the sun, moon, and stars were meticulously recorded, leading to accurate calendars that reflected these cycles. By correlating this astronomical data with their calendrical systems, the Aztecs could predict celestial events and seasons with remarkable accuracy, enabling them to plan agricultural activities and religious ceremonies accordingly. Such an understanding of astronomy was not simply for scientific pursuits; it was a means through which the Aztecs expressed their connection with the divine.

Aztec numerical and calendrical systems also had a social dimension. Mathematical knowledge and the ability to interpret the calendar were regarded as signs of status and education within Aztec society. Priests, scribes, and administrators were often among the most educated, interpreting the complex systems for the ruling elite and the broader population. This knowledge was considered sacred and was closely guarded, ensuring that the elites maintained control over religious practices and resource management through their expertise.

The arrival of the Spanish in the 16th century marked a significant turning point for Aztec mathematics and calendrical systems. The imposition of European calendars, religious practices, and a decimal numerical system brought profound changes to the Aztec ways of life. The new regime sought to dismantle traditional practices and beliefs in favor of their own, leading to the decline of the use of Aztec numerals and calendars. This cultural shift resulted in the widespread loss of indigenous knowledge, as the Spanish sought to replace native systems with their own methods of governance and record-keeping.

Though the majority of Aztec mathematical knowledge and practices were suppressed, modern scholars have made significant strides in reclaiming and understanding these ancient systems. Mesoamericanist researchers have studied various historical documents, archaeological sites, and codices to gain insight into the mathematical prowess of the Aztec civilization. These efforts have revealed that Aztec mathematics was not merely functional but nuanced, serving religious, social, and economic purposes that were essential for the survival and organization of the society.

Interdisciplinary studies incorporating anthropology, history, archaeology, and mathematics have broadened our understanding of the Aztecs' numerical and calendrical systems. The revival of traditional calendars among indigenous communities in contemporary Mexico has highlighted the enduring legacy of Aztec mathematics and its importance in cultural identity and resilience. For many, these ancient methods of timekeeping and calculation continue to play an integral part in communal life, agricultural practices, and ritual observances, proving that while the imposing forces of colonialism may have disrupted their original practices, the spirit of Aztec mathematics endures.

In conclusion, Aztec mathematics, characterized by a vigesimal numeral system and a profound calendrical framework, was central to the cultural and practical fabric of Aztec civilization. Through its links between mathematics, astronomy, agriculture, and spirituality, it facilitated not only administrative efficiency but also reinforced social and religious structures within society. Though the onset of colonialism dealt a significant blow to these ancient systems, the continuing interest in and revival of Mesoamerican knowledge today affirm the importance of understanding and preserving the complexities of Aztec mathematics for generations to come.